Journal of Mathematical Sciences, Vol. 74, No. 3, 1995

B E N D I N G OF S U R F A C E S . P A R T II I. Ivanova-Karatopraklieva and I. Kh. Sabitov

UDC 514.772.35

The first part of this work was published in 1991 [45]. According to the general scheme of the whole survey, outlined in the introduction (see [45]), the previous part was devoted to studies on bendings and infinitesimal bendings of surfaces in R a, with the surface classes set off with respect to the sign of the curvature. According to the same scheme, the part presented here is a review of studies on two special classes of surfaces: surfaces of revolution and polyhedrons. Section 7 is the result of the authors' joint work, while Section 8 was written by I. Kh. Sabitov alone. The references to the items of the first part of the survey are marked, for example, in the following way: i.l.2,I.

7.

BENDINGS

AND INFINITESIMAL

BENDINGS OF SURFACES OF REVOLUTION

The finding of infinitesimal bendings of surfaces of revolution is reduced to the solution of a certain system of ordinary differential equations. This leads to the fact that the surfaces of revolution occupy a remarkable position in the theory .of infinitesimal bendings because, in the first place, they represent a class of surfaces with a wide range of concrete properties of their infinitesimal bendings, in the second place, they are model surfaces which are used for verification of various hypotheses, and in the third place, they serve as a theoretical source of new hypotheses. 7.1.

E q u a t i o n of I n f i n i t e s i m a l B e n d i n g s of t h e S u r f a c e s o f R e v o l u t i o n

The main method of investigating infinitesimal bendings of the surfaces of revolution was introduced by S. Cohn-Vossen in 1929 (see [154]). This method implies that the position vector r of the surface of revolution S as well as the fields zj of the infinitesimal deformation of order n, 1 < j < n, are refered to the movable frame {k, e(0), e'(O)}, where k is a unit vector directed along the rotational axis Oz, e(6) = cos 0i+ sin 0j is a unit vector located perpendicularly to the axis Oz, which describes a circle under alteration of the angle Of revolution 0 from 0 to 27r; here i, j, k are the standard notations of basis vectors along the axes Ox, Oy, Oz. In the general case, when the equation of meridian L is thought to have a parametric representation z = ~(r),

p= r

(1)

(where p is the distance from the rotational axis), the position vector r is represented as

r(r,e) = ~(r)k + r

(2)

and the fields zj are sought in the following form: zS = as( ~, e)k + ~j(r, e)e(e) + ~ ( r , e)d(e),

1 < j _ n.

(3)

Since the equations for determining the field Zl of the infinitesimal deformation of order n are linear ones, by expansion of the components zx(r, 0) in the Fourier series =

dk)Cr) JR

'kt

=

= JR

(4) R

Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory. Vol. 8, Geometriya-1, 1993. 1072-3374/95/7403-0997512.50 9

Plenum Publishing Corporation

997

for a~~),

f~),

7~~) the following system of ordinary differential equations is obtained:

LI(,~ ~), ~)) _-__v'.~*)' + r

L~(4 ~), ZI*), 71~)) -= ~ v ' 4 ~) + r

= 0,

+ r

- 7~*)) = 0,

(the values k = O, • correspond to the trivial infinitesimal bendings). In the smoothness class C ~ the system (5) is reduced to the system of two equations

CZ~*)'+ ~2v'~ ~) + ( ~ - 1)r

~) =0,

Cv'.i~)'- ~v'r

r

~- 1)Z~) = 0,

~ _> ~

(6)

(taking into account that &~) -- a~-k), the same for f~) and 7~)), and in the smoothness class C 2 it can be reduced to one equation of second order: r162

*)" + r162

-2r

~)' - ~ r 1 6 2

- r

~) = 0

(7)

or

r

_ r

+ (~o'r - r

2 - 1)f~ ~) = 0.

(8)

The real solution of the system (6) or Eq. (7) or (8) for every k > 2 determines by the formulas (3)-(4) the field

z~*~ = [~k)(,)e'~~ + ~k~(r)e-'~~

+ [~*)(r)e" + ~)(,)e-'%(0)

+ [~)(,)~'~ + ~)(,)~-'%'(0)

(9)

r

of the infinitesimal bending of the first order, which is usually called the k-th fundamental harmonics or the k-th fundamental field of the first-order infinitesimal bending. Since the equations for determining the infinitesimal bendings superior to the first-order ones are not linear, the Fourier method without supplementary considerations can be applied for the extension of only those first-order infinitesimal bendings which consist of a finite number of fundamental harmonics. In particular, the fields zi, 2 < j < n, producing an extension of one harmonic z~k) into the infinitesimal bending of order n > 2, have coordinates

~(~, 0) = Z:[~*)(r)~ '~*~ + ~!~)(~)~-'~~ (p)

~(~, 0)= Z[~)(~)~

'~ + ~j~,)(~)e_,~,0],

(10)

(p)

"~j(T, O) = ~V'r'y(~%')e ,

+ ~Jv~)(r)e-'V*~

(P) on axes k,e,e ~ (where p = 0 , 2 , . . . , j for even j, and p = 1 , 3 , . . . , j for odd j). These coordinates are determined by solution of the inhomogeneous system of ordinary differential equations whose left members have the same form as in the above-mentioned system (5): L l ( a ? ~),/~J~)) = '

-~jlO(V~),

L2 (a~P~), a(~)-j,,~'(v*)~)= ~(~)~2,

Lz(/~Jv~),7J~)) = ~~z~(v~),

2 = 2 , ... ,n, (11)

and whose right members ~~(~) j~ , 1 = 1~2,3, depend on r,z~),zz,. 9 ,z~_~. In the smoothness class C 2, the solution of the system (11), for a given 2, can be reduced to one second-order inhomogeneous equation, written as

r 1 6 2" -c~j

+ r162

. 2r

.

k), . kZr162.

+ (r

~'r

r

Q!~k)

(12)

or

~,~p*),,-

r

2 - 1)z~p*) = R~~ .

(13)

The main difficulty in investigating systems (5) and (11) (or the corresponding equations of second order) is the presence of certain values of the parameter r, producing such singularities of the systems as vanishing of 998

J:: l

Fig. la coefficients under higher derivatives or a change of sign of certain coefficients, which has an essential influence on the properties of the solutions. The first singularity is obviously the pole, that is, the point where r = 0. Information about local study of the infinitesimal bendings in the pole neighborhood can be found in i.2.2,I, and here we remark only that e'/en a minimum requirement on the continuity of the field zl in the pole leads, as may be seen in (9), to the condition of vanishing of the functions a~~), ~/~k), ~/~k) at the point T = To. The parabolic parallels defined by the condition "curvature K = 0" constitute the second type of singularities of the systems. Assuming that the smoothness of the rotation 'curve L is of the class C 2 and its parametrization is a regular one, that is, opt2+ r ~ 0, let us give the classification of singular parallels according to Cohn-Vossen [154]. The following cases are possible: 1. Parabolic parallel of the first genus. On this parallel, the curvature of meridian L is equal to null and the tangent l to L is not perpendicular to the rotational axis, that is, at the corresponding point of the meridian ~o' # 0, ~'r - r = 0. Such parallels are shown in Figs. la-c. In ordinary cases, this parallel exerts no effect on the existence of infinitesimal bendings, but it has a pronounced impact on the local behavior of fields of infinitesimal bendings z ? ) when k ~ oo. 2. Parabolic parallel of the second genus. On this parallel, the curvature of meridian L is not equal to null and the tangent l to L is perpendicular to the rotational axis and has first-order tangency with L, that is, at the corresponding point of the meridian ~' = 0, ~" # 0 and, consequently, no neighborhood of this point is one-valued projected onto the rotational axis. Such a parallel is shown in Fig. 2. This point exerts a prominent influence on the first-order infinitesimal bendings, which are possible only under condition t h a t fl[k) = 0, 7~k) = 0. The second-order infinitesimal bendings are generally absent in their 9neighborhood (see i.7.7). 3. Parabolic parallel of the third genus. On this parallel, the curvature of meridian L is equal to null and the tangent I to L is perpendicular to the rotational axis and has m o r e than the first-order tangency with L, that is, at the corresponding point of the meridian ~' = 0, ~o" = 0. Such a parallel in the case of a meridian inflection point is shown in Fig. 3. This parallel also exerts a significant influence on the infinitesimal bendings of the first and second order (see i.7.7). 999

_/_

k J

Y

/

Fig. lb

)

Fig. lc An "exact" description of the surface structure in the neighborhood of singular parallels is possible only for the parabolic parallel of the second genus, in the sense that, in passing over this parallel, the surface curvature changes i'ts sign, while in approaching the parallels of the first and third genera the behavior of the surface curvature sign may be as arbitrary as is wished, for such parallels do not have to be isolated. It is worth noting that the points on parallels of the first and second genera are not the points of the surface flattening, while the parallel of the third genus throughout consists of the flattening points. For the study of the infinitesimal bendings in neighborhood of the pole and parabolic parallels of the second and third genera, usually the Fuchsian theory is used, with r = p taken as the parameter. In the neighborhood of the parallel of the first genus it is advantageous to take r = z. The parallels of the second and third genera are asymptotic curves of the surface in the sense that at each I000

Fig, 2

\

Y

Fig. 3 of their points the plane tangent to the surface is coincident with the plane of the parallel and it intersects the surface in this very parallel. As mentioned above, these parallels place a priori certain constraints on the behavior of the bending fields, thus imposing additional limits on the capability of the surface to be flexible. A similar fact was also established in [14] in a more general situation, and not just for the surface of revolution.

7.2.

Closed R e g u l a r Surfaces of R e v o l u t i o n

In the work published by N. Liebmann in 1900 [188], the infinitesimal rigidity and analytical-by-parameter inflexibility of ovaloids with strictly positive curvature (in the class of analytical surfaces) was established. Later on, Liebmann himself raised the question - - is it of principle importance that in this case the surface contains no points with negative curvature K - - and he provided the answer to this question by proving [189] that the circular torus having "one-half" of its surface comprised of points with negative curvature is also I001

infinitesimally rigid and, as a consequence, inflexible (in the class of deformations analytical by parameter), a After this it was reasonable that the question arose as to whether there is at all a closed nonrigid surface in existence. In 1929, S. Cohn-Vossen [154] gave a positive answer to this question. He showed the existence of infinitesimally nonrigid dosed regular surfaces of revolution of both possible topological types: sphere and toms. In the same work he presented a condition of continuation of the first-order infinitesimal bending into the second-order infinitesimal bending for the sphere-type surface of revolution (without asymptotic parallels). For this it is sufficient that the field of first-order infinitesimal bending of the surface 5' should have only one fundamental harmonic. Thus, the question about the number of fundamental fields of first-order infinitesimal bending received an additional new content, in the above-mentioned paper [154], Cohn-Vossen also posed the question about the existence of the infinitesimal bending of order 3 and over. The first answers to the questions raised by Cohn-Vossen were obtained only thirty years later. In papers by B. A. Bublik ([18] and [19]) examples of surfaces of revolution with at least two linearly independent first-order infinitesimal bendings were constructed (but the precise number of fundamental harmonics was not determined in these works). Yu. G. Reshetnyak [103] pointed out a method for constructing the surfaces of revolution of the U ~ class with a finite number of fields of infinitesimal bendings having precisely preassigned numbers of harmonics kl < ... < km < ~ . In the paper by D. A. Trotsenko [125] the idea of the method presented by Reshetnyak in [103] was applied to the construction of an analytical surface of revolution with similar properties of its infinitesimal bendings (the main difficulty in constructing such examples lies in the following: the continuation of the solution of Eq. (8), which is regular in one pole, as a rule, appears to be unbounded in the other pole, and, consequently, it is necessary to "glue together" t h e solutions with one number k, singly regular in each of these poles). But so far it is a puzzle as to whether there exist regular surfaces of revolution with a complete (i.e., for all k _> 2) or at least, in general, an infinite system of regular fundamental harmonics. Such an example is known in the irregular case: E. G. Poznyak [94] has constructed in an explicit form a closed surface of revolution which is regular everywhere except at one pole of the peak type and which has the complete system of regular fundamental harmonics. In all works where examples of infinitesimally nonrigid regular surfaces of revolution were constructed, the curvature in the surface poles was positive. Most likely, this fact is not a coincidental one but is the result of the essence of the matter, at least in the analytical class: if the pole of an analytic surface of revolution is the flattening point, then, as has been proved in [110], there does not exist infinitesimal bending in the C~-regularity class, even locally in the neighborhood of such a pole or completely (when the pole has the 8th order of flattening, that is, z = ~v(p) ,,~ plo), or there exists a finite n u m b e r of such bendings (the surface poles have the following form: z = ~(p) ~ p~(2,,~+2,~+1), ra > 2), or the numbers of the non-trivial fundamental harmonics are determined according to a certain Diophantine equation and follow with large gaps. Consequently, if two poles of the dosed surface have different flattening orders with noncoincidental solutions of the above-mentioned Diophantine equation, then the integrals of Eq. (8), which are regular in one pole, are certain to be irregular in another pole and therefore such analytic surfaces of revolution will be infinitesimally rigid in Ca-class and thus inflexible with respect to analytic-by-parameter deformations. It seems likely that the neighborhoods of such poles have second-order rigidity for almost all flattening orders (in the case of C~-class infinitesimal bendings). In [57], A. D. Milka has indicated a method for constructing infinitesimally nonrigid closed surfaces of revolution, in which..Eq. (7) is considered as an equation related to the meridian with a certain preassigned regular function a~ k). In this case the surface, as well as one of its bending fields, can be obtained in an explicit form, but the problem of the existence or absence of other bending fields still remains unsolved. Cohn-Vossen's studies on the extension of the first-order infinitesimal bending to the second-order infinitesimal bending were borne out by the works of E. G. Poznyak. It was revealed that if for the regular surface of revolution (without an asymptotic parallel) with a meridian, p = p ( z ) , a < z < b, and nonparabolic 1 Other proofs of the circular torus infinitesimal rigidity are available, see, for example, [2, 193, 54].

1002

poles z = a and z = b Eq. (8) is regularly solvable for a given k _> 2 and unsolvable for 2k, then the firstorder infinitesimal bending with fundamental harmonic z[k) may be regularly extended to the second-order infinitesimal bending [95]. The existence of surfaces with a finite number of linearly independent first-order infinitesimal bendings was also demonstrated [97] (here the results obtained at a later time in the works of Yu. G. Reshetnyak [103] and D. A. Trotsenko [125] can also be used. Thus, the existence of regular closed surfaces of revolution with second-order nonrigidity has been shown. Under certain additional conditions on the fields of bending along a given parallel, such surfaces appear to be rigid with respect to the second-order infinitesimal bendings [42]. The case of parabolic poles has been studied in [44]. The general result of this work is that the higher the order of flattening at the pole, the less the possibility of continuation of the harmonic z~k) to the second-order infinitesimal bending. The investigations of the higher-order infinitesimal bendings of sphere-type closed regular surfaces started quite recently, and here there are only a few results. In i.7.7 we shall see that the presence of a parabolic parallel of the second or third genus on the surface leads to the second-order rigidity; therefore the infinitesimal bendings of the higher orders are considered only for the surfaces without such parallels. In the paper by I. Ivanova-Karatopraklieva [182], regular surfaces with poles of both kinds - - with flattening and without flattening - - have been studied. The conditions of extension of the field of the first-order infinitesimal bending consisting of one fundamental harmonic z[ I') to the infinitesimal bending of order n > 3 were also obtained. At first some necessary and sufficient conditions for local extension were given, which take into account the number of the harmonic k, the order of the infinitesimal bending n, and the order of the flattening. Then some sufficient conditions for,the rigidity or flexibility of the nth order were specified, related mainly to the suppositions on the existence or absence of the first-order infinitesimal bending with definite numbers of harmonics. It should be noted, however, that these conditions are rather difficult to verify because the real existence of an example of a regfllar closed nonrigid surface (even of the third order) has not been recognized yet. As may be seen from the above-mentioned review, the main published results are concerned with the investigation of cases of the first- and second-order nonrigidity. There has not been obtained any general sufficient or necessary condition for the rigidity of the surfaces nonconvex as a whole, except for a longknown criterion of the first-order rigidity in the presence of a convex belt with spherical image area equal to 47r and one new criterion relating to analytic infinitesimal bendings of analytic surfaces with two poles of flattening [110]. This fact is explicable on the basis of the following Cohn-Vossen's study results [154]: in the metric U 1, the closed nonrigid surfaces of revolution are distributed everywhere dense among the rigid ones, including even the surfaces with positive curvature. There are no general assertions on the distribution of nonrigid surfaces among the rigid ones in the metric U 2, but the example of E. Rembs [197] with the family of varied Bernoulli's lemniscates and the example of a more general single-parameter family of regular surfaces of revolution with alternating curvature, presented by Ivanova-Karatopraklieva in [38], indicate that in these families the nonrigid surfaces form a countable set with convergence to accumulation points in the metric U 2. The existence of infinitesimally flexible closed surfaces of revolution naturally leads to the question about the effect of deformation on the volume of a body bounded by a nonrigid surface. This problem was solved by V. A. Alexandrov [6]. It turns out that the,volume of a body of rotation with a nonrigid surface devoid of flat pieces is stationary under infinitesimal bending of the body boundary (see also [30]). In this case the surfaces are supposed to be Cl-smooth and of both possible types - - t o r u s or sphere (for sphere, irregular poles are admitted). We have no information about any theorem concerning the bending of closed surfaces of revolution. It has been established under the additional requirement of preserving the mean curvature [71] that if the surface of revolution is star-shaped with respect to a certain point, it allows no isometric deformation. The problem of bending and infinitesimal bending of surfaces which are locally the surfaces of revolution has not yet been studied; note that such surfaces may have any topological type [109]. Among the problems concerning the characteristics of the intrinsic metric of the surfaces of revolution, the following problem still

1003

k

_

Fig. 4a

Fig. 4b remains unsolved (see, for example, [209]): if the compact surface of the genus null or one admits bending on itself, that is, all the surface points remain on the surface in the process of its bending, then the metric of such a surface is the metric of a surface of revolution. 7.3.

P i e c e w i s e R e g u l a r Closed Surfaces of R e v o l u t i o n

If the rotation meridian L is a piecewise regular curve, then its rotation produces a piecewise regular surface on which thdedges will be parallels corresponding to points of the joint of the smooth arcs of the curve L. Since the formal part of the classical theory on infinitesimal bendings supposes at least C 1 smoothness of the surface and the field of its infinitesimal bendings, the determination and study of infinitesimal bendings in the not-smooth case require some additional investigations. Such investigations have been performed by A. D. Alexandrov [1] for the first-order bendings of convex surfaces (more detailed information on this problem can be found in i.3.3, I). In the same paper, he has proved that every convex surface of revolution devoid of fiat pieces, whose spherical image area is equal to 41r, is infinitesimally rigid and all other convex surfaces of revolution axe nonrigid. It is pertinent to note that there exists no theory of second- and higher-order infinitesimal bendings for general convex surfaces, constructed with the same maximum generality as the 1004

Zt

Z2 Fig, 5a

].--

Fig. 5b theory for cases of first-order bendings. We shall consider other works associated with infinitesimal bendings of the general convex surfaces of revolution later. Now we shall suppose that the surface regularity is broken either in poles (the poles are conic or have the form of a peak) or on a finite number of parallels. Then, on the separate regular pieces, including edges as the boundaries, the infinitesimal bendings will be understood in the usual way, but on the surface as a whole the field of infinitesimal bendings is supposed to be continuous only. This requirement leads to certain conditions for the behavior of infinitesimal bendings in approaching irregular poles or edges. A typical example is the condition imposed on the field of revolution y (information on it can be found in i.1.5, I): on the line of a smooth edge, the difference y+ - y - should be parallel with the tangent to the edge, where y* are the limits of y in approaching the edge line on various sides. The book by I. N. Vekua [23] provided a detailed analysis of the behavior of various vector and scalar fields associated with fields of infinitesimal bendings on the edges. One of the first results of the investigation of the infinitesimal bendings of closed piecewise convex surfaces of revolution has been obtained in work of B. V. Boyarskii [16], where a certain integral formula was 1005

Fig. 5c successfully used to prove the infinitesimal rigidity of the surface produced by the rotation of convex curves whose form is represented in Fig. 4a and 4b. It is worth noting that if the surfaces of the indicated type have self-intersection, then it is easy to verify on examples that they can already be nonrigid. The work by A. D. Milka [57] indicates a method for constructing an infinitesimally nonrigid closed surface made up of two pieces of various rotational ovaloids glued together internally when the line of gluing is the contact (tangent) line of the surfaces. The paper of I. Kh. Sabitov [104] generalizes the results obtained by B. V. Boyarskii. The possible types of mutual arrangement of the convex arcs, whose rotation produces the piecewise regular closed surfaces described in [104], are presented in Figs. 5a-bc. In the same paper ([104]) a series of effectively verifiable sufficient conditions of the infinitesimal rigidity of such surfaces was proposed (for example, if the domain of representation of each below-placed arc presented in Fig. 5a is limited by the boundaries of the representation domain of the above-placed arc, the surface is infinitesimally rigid). The idea of the proof generalizes the well-known reasoning which gives the infinitesimal rigidity of the closed convex surface of revolution: the solution/3~(z) of the equation p(z)3'k'(z) + (k 2

-

1)p"(z)flk(z) = O,

zl < z < z2,

(14)

(resulting from (8) with r = z, p = p(z)), which vanishes at the point of the first pole z = zl, remains positive for z > zl and is not equal to zero at the second pole z = z2. But since the curves p~(z) participating in the formation of the surfaces of revolution studied in [104] have various domains of definition in the general case, the conjugation of solutions ~,i of the corresponding equations (140 by the condition of continuity of the infinitesimal bending field and tracing of the change of their sign from one pole to another take place under nonstandard conditions. In [16] and [104], every two adjacent (not strictly) convex surfaces of revolution are glued together internally (about internal and external gluing, see i.4.2, I). If two convex surfaces of revolution are glued together externally, thus producing a nonconvex surface as a whole, a nonrigid closed surface may already be obtained. A well-known classical example (see [23]) is the external gluing together of two similar spherical segments, greater than hemispheres and bounded by Liebmann's parallels (for information on them, see i.7.4). As a generalization of this example, B. V. Boyarsky (see [23]) has given the necessary and sufficient conditions of nonrigidity of the closed surface formed by external gluing together of two spherical segments with different radii. The case of gluing together of pieces of second-degree surfaces with positive curvature was studied by Z. D. Usmanov [128] using Vekua's method. There are also works in which, among the pieces glued together, there is always a surface of null curvature: a cylindrical or conical zone. A case of external gluing together of two spherical segments by a cylindrical zone has been investigated in [116]. Various combinations of external and internal gluing together of two segments of second-degree surfaces of revolution (ellipsoid and paraboloid) with the help of a cylindrical or conical zone were studied by V. I. Mikhailovskii in [66]. 1006

kFig. 6 Finally, in the works of M. N. Radchenko [99] and I. Kh. Sabitov [104], the infinitesimal rigidity of the closed piecewise regular torus-like surfaces of revolution with meridian similar to the one pictured in Fig. 6 was proved (in [99] there is the addi'tional condition that the convex meridian arcs have a common tangent perpendicular to the axis of revolution in one of two points of the joint). Since, among the piecewise convex closed surfaces of revolution, one also encounters infinitesimally flexible ones, the problem arises of Studying the distribution of nonrigid surfaces among all surfaces of the abovementioned class (as in the regular ease). One of the general results in this direction has been obtained in paper by V. T. Fomenko [129]: (1) every regular closed surface of revolution S with positive curvature is the limit one for a set of infinitesimally nonrigid closed non convex as a whole surfaces of revolution S~, formed by the external gluing together of two regular pieces along any fixed parallel of the surface 5'; (2) every nonrigid piecewise convex surface of revolution of S, type may be included in a continuous family of nonrigid and projectively unequivalent surfaces of the same type. I. Ivanova-Karatopraklieva [31] has studied the dosed surfaces of revolution Ex whose meridian is obtained from the meridian displayed in Fig. 5a by substitution of the last arc by a piecewise convex arc Ln U L~, where the rotation of the convex curves L , : p = p,(z) > 0,

L~: p = p,+l(~ + ( z - ~)/~),

~ > 0,

produces the surfaces S~ C Ex and Sx C Ea which are externally glued along their common parallel z = ~. (the pole belongs to the surface Sx). It has been demonstrated that among the surfaces 5In exists a precisely countable set of infinitesimally nonrigid surfaces if ' > 0. which condense to the surface Sa0, )~o = Pn+I( )/Pn( ) smooth along the parallel z = ~, and the class of surfaces Ea contains, at most, a countable number of nonrigid surfaces if p'+l(~)/p'(~.) < 0. It should be noted that for n = 1 this result was obtained by B. V. Boyarskii (see [23]) on the basis of a Rembs estimate of the ratio 13'k(z)/t3k(z ) with k --* +oo. The so-called ribbed surfaces, whose meridian is a polygonal line with a finite number of rectilinear links, occupy an important place among the closed piecewise regular surfaces of revolution. Infinitesimal bending fields of the surface of revolution of each link can be described in an explicit form, which is why such surfaces are usually used for constructing various interesting examples. Even S. Cohn-Vossen [154] has resorted to this property of the above-mentioned fields. In the class of closed surfaces of the null genus, the main consideration is given to meridians with a single-valued projection on the axis of revolution, that is, the case of consecutive external gluing together of the conical and cylindrical belts of revolution. The main objects of investigation are the same as in the cases of other piecewise continuous surfaces. 1007

B. A. Bublik has proved (see [20]) that a closed ribbed surface with meridian formed by (n + 2)-links allows no more than n fundamental fields of first-order infinitesimal bending. He has constructed examples with strictly two or three linearly independent fields of infinitesimal bendings. Examples of ribbed surfaces with two independent fields of infinitesimal bendings are also presented in the paper by V. I. Shimko [137]. The works of N. G. Perlova [77, 78, 84, 86] are devoted to the extension of the first-order infinitesimal bendings to the infinitesimal bendings of second and higher order. In particular, these works display sufficient conditions of the extendibility of the m-order infinitesimal bendings to the infinitesimal bending of the following order (m + 1) (in the case of an arbitrary number of links). Examples of surfaces with infinitesimal rigidity and nonrigidity of the second and third orders are constructed in these papers. It should be stressed that these examples are the only known examples of closed surfaces with third-order nonrigidity; thus, we shall cite one of them: a corresponding polygonal line consists of four links with vertexes in the points 3(1 (z,p) = (0,01,(1,1), (1 +a2, ~ _ --~ a2)), (1 -4- 2a2, 1), (2 A- 2a2, 0) where a2 = - l / a 2 ( 1 - 3a2) and a2, is a root of the equation 63a~ - 66a~ + 27a22 - 36a2 - 64 = 0 with the condition - 1 < c~2 < 0; rotation is about the axis Oz (example from [77] with regard for the correction made in [82]). Torus-like ribbed surfaces of revolution have been scantily studied; here we can mention the work of K. M. Belov [15], where the necessary and Sufficient condition of infinitesimal nonrigidity is given for the torus with quadrangular meridian in a special position (in particular, there exist nonrigid tori with a convex quadrangular meridian). In her paper [85], N. G. Perlova gave the necessary and sufficient conditions of the nonrigidity of a torus of revolution with meridian consisting of any finite number of links. We shall end the review of the infinitesimal bendings of closed ribbed surfaces with the following problem. The extension of infinitesimal bendings of lower orders to the infinitesimal bendings of higher orders requires additional conditions at every step. Can it be advocated that for the number n of links of the rotational polygonal line there exists an integer N ( n ) such that the corresponding ribbed surface may have infinitesimal bendings only of order not higher than N(n)? Or, if it is not true, may such a surface admit analytlcal-byparameter bendings? 7.4.

N o n - N e g a t i v e C u r v a t u r e Surfaces w i t h B o u n d a r y

If any arbitrarily small neighborhood of the pole is cut out of a closed surface of revolution of the null genus not containing an asymptotic parallel, then the remaining part of the surface will become infinitesimally nonrigid with respect to infinitesimal bendings of any order [182]. Because of this, the investigations of bendings and infinitesimal bendings of surfaces of revolution with a boundary involve, as a rule, some boundary conditions restricting the admissible deformations of surfaces. It is desirable that these conditions possess geometrically comprehensible content (about the possible types of boundary conditions, see i.l.8,I) Conical s o c k e t c o n n e c t i o n s . In cases of such connections, it is supposed that the boundary L of a surface lies on a certain cone (socket) and in the process of deformation it slides over the surface of this cone, thus leading to boundary condition of the type Un = 0 for the infinitesimal bendings, where U is the field of infinitesimal bendings of a given order and n is the normal to the conical socket. The following simple case is most frequently studied: the boundary of the surface is a parallel (one or two), and the apex of the cone is located on the axis of revolution of the surface. The method of investigation calls for studying the spectrum of the boundary-value problem for the equations and systems of the corresponding type. The first works devoted to this theme were the publications of I. N. Vekua, which were partially repeated in [23]. In this work [23] and in the paper by Sun He-Sheng [115], the rigidity of a convex regular surface bounded by one or two parallels with conical socket connections laid on the boundaries is studied. Besides proving the general properties about the countable character of the distribution of the nonrigid socket connections 1008

condensing to the orthogonal socket (which turns out to be a rigid one), these papers present a precise picture of the arrangement of nonrigid conical sockets for concrete surfaces: ellipsoid, sphere, etc. A more general case is considered in [129] and [130], where the normal to a conical socket forms an nonconstant angle with the normal to the surface. It turns out that in this case the set of nonrigid conical socket connections has the power of the continuum, and, what is more, the nonrigid socket connections are arranged everywhere dense in the neighborhood of the orthogonal socket. Additional conditions (the fixation of certain points of the boundary according to a special lawof their arrangement) providing the surface rigidity in case of nonrigid socket connection are indicated in [134]. For a closed Cl-smooth convex surface, the existence of a countable number of analogs of Liebmann's parallels in relation to a socket which has the form of a cone coaxial with the surface is proved in [7]. Information on this problem can also be found in [69]. The conical socket connections have also been studied for the piecewise convex surfaces with meridian shown in Fig. 5a. This case is investigated in [8, 10], which describe the arrangement of parallels limiting the surface admitting nontrivial infinitesimal bendings of the first, second, and third orders of sliding over the conical socket. S l i d i n g r e l a t i v e t o p l a n e s . The main classical result concerning this problem dates back to N. Liebmann and E. Rembs. Rembs generalized the result obtained in [190] for a sphere and showed in [198] that on every closed regular surface of revolution with positive curvature there exists a countable set of parallels Lk such t h a t the greatest (by the area of a spherical image) part of the surface bounded by the parallel Lk admits infinitesimal bendings of the. first-order sliding relative to the plane of the parallel L~ itself. These parallels, called Liebmann's parallels, accumulate at the largest parallel L0 of the surface (analog of a sphere equator), but none of the surface parts with boundary L0 admits infinitesimal bendings of sliding. Rembs [199] has demonstrated the presence of the second-order sliding infinitesimal bending on the same parallels Lk (but the third-order sliding bendings are absent - - it is shown for a sphere). This result obtained by Rembs has been generalized in the following way. In the first place, the conditions of regularity were weakened down to the minimum possible one in [12]: the Rembs theorem on the first-order bendings was proved (with the same assertion) for the general convex surfaces of revolution. In the second place, the planes of sliding were investigated in [57] in an arbitrary arrangement, and the existence of the corresponding analogs of Liebmann's parallels was also shown for them (for a sphere, see [132]). The case of infinite convex surfaces of revolution is studied in [98]. The infinitesimal bendings of the generalized sliding are investigated for concrete surfaces in [46] and [47]. For a closed surface of the class C 2, the existence of a countable set of zones bounded by parallels and admittance of the infinitesimal bendings of sliding of the first and second orders along the boundary planes is proved in [87]. Infinitesimal bendings of sliding of the first and second orders in the piecewise convex case are described in [32, 36, 41] for the surfaces with meridians shown in Fig. 5a. Conditions for the arrangement of the penultimate arc p,~_l(z) are indicated (in dependence on the parity n), the fulfillment of which at the last piece S,, (produced by rotation of the arc p,,(z)) provides the existence of a countable set of Liebmann's parallels Lk C S,,, k = 1, 2,..., accumulating at the maximal parallel of the surface S,~. For the same surfaces, the infinitesimal bendings of sliding of the third a n d higher orders are studied in [9] and [42]. The essence of reasoning in all works devoted to Liebmann's parallels and their analogs is the elucidation of the behavior of the ~'elation ~'k/~k(z) as k ~ oo (fi,(z) is the solution of Eq. (14)). For this relation, in [12] a certain part of Rembs results from [198] is transferred to the case of general convex surfaces of revolution; in [31] this relation is studied in the case of piecewise convex surfaces. F i x a t i o n r e l a t i v e t o a p o i n t . It will be recalled (see i.1.8,I) that if the distances from points of the boundary of a deformed surface to a certain given point M are stationary at the initial moment of nontrivial bending, then such a point M is called the point of relative nonrigidity. If the convex surface S is starshaped relative to a point M and a dividing plane can be inserted between S and M, then the fixation S relative to the point M leads to inflexibility and infinitesimal rigidity (see i.3.2,I and 3.3,I). A. D. Milka [57] 1009

has investigated the distribution of points of relative infinitesimal nonrigidity in R 3 for the simply connected regular convex surface of revolution with boundary coinciding with a parallel and, in particular, he has proved that in a certain part of R 3, such points fill up the countable set of analytical surfaces with a common border (for example, for the hemisphere z = ~/1 - x 2 - y2 such surfaces have the equations z = nx/1 - x 2 - y2, n = 2, 3,...). The part of the space R 3 in which the points of relative nonrigidity are afortiori absent has also been indicated, but up till now a complete description of the set of points of relative nonrigidity has not been performed. For piecewise convex surfaces, in [8, 11, 43] the distribution of points of relative nonrigidity of the first, second, and third orders is studied in cases where the points are arranged on the axis of revolution. It is shown that if the boundary of the surface with meridian shown in Fig. 5a is the parallel on the last piece S,, then there exists a countable set of points of relative nonrigidity of the first and second order on the axis of revolution, which accumulate at a definite point. O t h e r b o u n d a r y c o n d i t i o n s . A number of works are dedicated to infinitesimal bendings with definite requirements on the behavior at the boundary of various geometrical characteristics - - in the case of convex surfaces, these are the works [51], [75] and in the case of piecewise convex surfaces, works [33] and [36]. A physical interpretation of certain values associated with the infinitesimal bendings of surfaces of revolution with positive curvature through stresses and tensions in the corresponding membrane theory of elastic shells is given in [49, 50]. D. Bleecker in [149] describes in an explicit form the known fundamental harmonics for infinitesimal bendings of a sphere with one removed pole, which serve as the basis for an analog of Poisson's formula for calculating the field of an infinitesimal bending of a spherical segment under the given value of one of the field components at the boundary of the segment. Bendings and isometric deformations of surfaces of revolution with a boundary have practically not been investigated. We know of on!y one work [194] where the isometric deformations of a spherical zone are considered under the condition that the normal curvatures pl and p2 of the boundary of the deformed surface are constant along the boundary. It is shown that if the boundaries of a spherical zone have the same radii R1 = R~, then, when pl # p2, there exist no surfaces nontrivially isometric to the spherical zone, and when pl = p2, there exists a single-parametric family of such surfaces. If R1 # R2, then under the condition A = [R 2 (p2- 1 ) -

-

> 0

there exist precisely two surfaces which are isometrical to the zone, and for A < 0 none of these surfaces exist. 7.5.

Nonpositive Curvature Surfaces with a Boundary

Since a regular surface of revolution cannot have nonpositive curvature in the neighborhood of a pole, the doubly connected surfaces of revolution bounded by two parallels are usually investigated in this class of curvature. The following two main problems are studied in the theory of infinitesimal bendings of such surfaces. A. A certain parallel Lo is given on the surface S (without asymptotic parallels) and such parallels L, : z = z, are searched for which the zone S(") C S, bounded by parallels Lo and L~, should allow the infinitesimal bendings of a given order satisfying some boundary conditions imposed beforehand on L0 and L,. If such parallels L, exist, an effort is usually made to determine the power of the set of such parallels with an indication of the law of their arrangement among other parallels. B. In other serles of problems, the boundary parallels and their planes are fixed, but the surface of revolution stretched over them varies. An example: the surfaces S,~ under consideration have meridian

p(z)=h+~(z),

~(z)>O,

~(z) e C2[a, hi,

h e [0,+oo);

the problem of finding those of their infinitesimal bendings for which some boundary conditions are imposed along the boundary parallels L, : z = a and Lb : z = b is solved. If nonrigid surfaces occur under these conditions, the power of their set and the law of distribution among all surfaces Sh are studied. The problem A has been investigated in [115, 118, 61-65, 67, 131, 89]. The problem of finding infinitesimal bendings with socket connections at the boundary as for zones of a one-sheeted hyperboloid of revolution is 1010

solved in [115, 118]. The same problem has been studied in [63] and [65] for belts of C 2 piecewise-smooth surfaces of revolution. The infinitesimal bendings of C 2 piecewise-smooth zones S ('~) C S, where the boundary parallels Lo and L,~ slide in their own planes, are studied in [62, 64]. The conditions of infinitesimal nonrigidity of the following surfaces glued with flat pieces along the parallels L0 and L~ are studied in [61]: an exponential tube with meridian

p = Aa ;~z,

a > O,

a ~ 1,

A > O,

A = const ~ 0 ,

a catenoid, a one-sheeted hyperboloid of revolution, and a paraboloid tube with meridian

p = A z '~,

A>0,

a=

const,

a<0

or

a>l.

The same boundary problem has been studied in [62] for the zones S(") on a C 2 smooth surface of revolution S. The infinitesimal bendings of C 2 smooth zones S ('~) C S, where L0 and L,, are fixed relative to two points or a plane parallel to the axis of revolution, have been investigated in [68], as well as in cases of restrictions allowing transformation of points along one of the boundary parallels in the predetermined direction. Infinitesimal bendings preserving certain geometrical values along L0 and Ln as well as under certain other special conditions on fields of bending along the boundary have been studied in [131, 89]. The results obtained in all these works pertain mainly to the proof of the existence o f a countable everywhere dense set of parallels {L~} on the surface S, such that the zones S ('~) are nonrigid. In addition, in [115] and [118] the minimal arc of the parallel L~ whose fixation results in the rigidity of the zone S (~) on a one-sheeted hyperboloid of revolution has been found. The problem of type B has been first investigated by Rembs [197] for the case in which the boundary parallels L~ and Lb are asymptotic ones. Such a zone has a spherical image 4a', but it appears that the zone of negative curvature may be nonrigid unlike the convex zone with the same boundary parallels. Rembs has shown that, among the surfaces Sh, there is a countable set of nonrigid ones. Moreover, it has been determined in [246] that their set is everywhere dense in Sh; the case of a meridian containing an arc which has the form of a polygonal line is presented in [207]. The problem B has been studied in [62-64] for piecewise C 2 smooth zones Sh (without asymptotic parallels) under boundary conditions of sliding along the border relative to a plane or under socket connections along the edge, as well as for C 2 smooth glued zones Sh. It has been proved that in all cases there exists a countable everywhere dense set of nonrigid surfaces among the surfaces Sh. Various concrete problems different from problems A and B have been studied in some papers. For instance, the rigidity of a simply connected domain bounded by two asymptotic curves and an arc AB of a parabolic parallel of the second genus under the condition of arc AB fixation has been proved in [215]. The infinitesimal bendings of a simply connected piece of a one-sheeted hyperboloid of revolution under certain special conditions on the field of bending along the boundary (the boundary is comprised of two curves, one of which is the arc of the parallel) are investigated in [91], with application of the known results of the theory of systems of differential equations. Infinitesimal bendings of noncompact surfaces of revolution with negative curvature have been studied in [100, 101, 25]. The behavior of field z of the first-order infinitesimal bending of a simply connected infinite negative curvature surface of revolution with the pole in the form of a peak at infinity has been studied in [100, 101]. R. F. Galeeva and D. D. Sokolov have demonstrated [25] that a one-sheeted hyperboloid of revolution is infinitesimally rigid in the class of bending fields converging to zero at infinity. Finally, we point out the paper of P. I. Kudrik [50], where the infinitesimal bendings of surfaces with negative curvature are investigated with the help of a p-hyperbolic system of differential equations.

7.6.

Zero Curvature Surfaces with Boundary

As has been shown in i.7.3, by such surfaces we mean surfaces produced by rotation of a polygonal line with rectilinear links, which are also called ribbed surfaces. The pole of such surfaces, when present, is 1011

always conical. The range of problems for these surfaces is nearly the same as for surfaces with nonnegative curvature; problem A is also encountered as for surfaces of nonpositive curvature. First of all, we shall mention the works [74, 76, 79], where Liebmann's parallels are searched for simply connected and doubly connected ribbed surfaces. The results obtained in these studies are similar to the Rembs case. Slidings over the cone with a given angle at the apex, which is coaxial with the surface, are studied in [133, 135], where conditions of existence of the parallels intercepting from the surface the parts which are nonrigid relative to this socket connection are found. It is also demonstrated that such parallels on a given link may comprise a countable set with only one point of accumulation. The following series of results refers to the extension of infinitesimal bendings of sliding of the first order to infinitesimal bendings of sliding of the second and third order: [74, 76, 82, 90]. In particular, in [82] it is shown that a simply connected ribbed surface admitting infinitesimal bendings of sliding of the first order over the coaxial cone or over the plane admits also an infinitesimal bending of sliding of the second order belonging to the same type, but this surface is rigid relative to the infinitesimal bendings of the second order if the boundary parallel slides over the surface of the coaxial cylinder. The conditions of extendibility of the above-mentioned infinitesimal bendings to infinitesimal bendings of the third order, satisfying the same conditions, have been also indicated. Concrete examples of ribbed surfaces admitting infinitesimal bendings of sliding of the third order over the plane of the boundary can be found in [82, 90]. In [80] it has been demonstrated that a ribbed trough of revolution possesses nonrigidity of any finite order. The infinitesimal bendings under the condition of boundary fixation relative to a point were investigated in [84] and [67]. :" The geometrical conditions on the deformations of a surface boundary are studied in [81]. It has been shown in this paper that a ribbed surface admits no infinitesimal bendings of the second order under the condition of presentation of the boundary's normal curvature. But there exist ribbed surfaces possessing nonrigidity of second order under the condition of preservation of the boundary's geodesic torsion. Finally, we mention the paper [102], where some properties of displacement fields and rotation fields of infinitesimal bendings of a right circular cylinder are studied. 7.7.

Surfaces of A l t e r n a t i n g C u r v a t u r e

For regular surfaces of alternating curvature the corresponding equations of infinitesimal bendings belong to equations of mixed type. I. N. Vekua [23] pointed out the significance of studying such equations just in relation to problems of the theory of infinitesimal bendings. If the change of sign in the surface curvature is induced by the presence of a parabolic parallel of the first genus on the surface, attempts to obtain any specific consequences from this property of the surface fail. However, if there are parabolic parallels of the second and third genus on the surface, a wide range of various results is obtained with the weakest suppositions on the smoothness of the surface and the field of its infinitesimal bendings. The starting result in these studies was the following assertion: L e m m a (T. M i n a g a w a - W. Rado [193]). Let the meridian L:

z=qa(v),

p=r

r e [0, a]

of the CLsmooth surface of revolution satisfy the following conditions: 1) ~ ' ( r ) ~ 0 , r a); 2) r 0 and if ~'(0) = 0, then there exists 5, 0 < 5 < a, such that [~o'(r)[ is nondecreasing in [0,5]. Then, if the parallel r = 0 is fixed, the surface S is rigid. This lemma allows one, for example, to obtain the infinitesimal rigidity of the surface of revolution S (without boundary or with boundary along the parallel; and the case of a more general boundary, see below) under the following requirements: S contains an open convex (in the general sense, that is, without any requirement of smoothness) zone 5'0 (without flat pieces) with complete spherical image 4~r, and part of the surface S - So is supposed to be CLsmooth, while the set of points in which the tangent to S is perpendicular, 1012

to the axis of revolution is isolated, and the conditions of the lemma concerning the structure of the surface are fulfilled in each neighborhood of such a point. Indeed, the convex zone So with complete spherical image 4z" is " infinitesimally rigid (see i.7.2), the field of infinitesimal bending zl at its boundary can be considered as equal to zero, and the continuation of the solution of the Cauchy problem for system (6) with zero initial conditions may be hindered only by alternately encountered points where ~'(z) = 0, but at t h e m by continuity (from the left or from the right) zl = 0, and through t h e m the equality zl = 0 is extended according to the lemma of Minagawa-Rado. In general, it is sufficient to have rigidity of any domain containing a complete parallel (for example, as the result of a "bad" pole) and to demand the fulfillment of the conditions of the Minagawa-Rado lemma for the remaining parts of the surface for the rigidity of the whole surface to be obtained. By virtue of importance of this lemma, there arises the question on the possibility of weakening the conditions of the lemma for the meridian of a surface of revolution. This question has been solved in [107], where a series of simple sufficient (but wider than in [193]) conditions are given on the basis of a certain criterion for the meridian z = ~(p) E C 1 with = 0,

r 0,

0 < Ip - p01 <

These conditions provide the validity of the lemma (for instance, for this purpose it will suffice that the function g(p) =

r

be integrable from the viewpoint o f Riemann's improper integral). Moreover, an example demonstrates that there exist smooth surfaces for which the fixation of the parallel with ~'(po) = 0 does not induce triviality of the field of infinitesimal bendings in the neighborhood of this parallel. It has been noted in [34] that under the conditions of the Minagawa-Rado lemma the statement about the triviality of infinitesimal bendings in the neighborhood of the parabolic parallel p = p0 is true also in the case in which the fixation of this parallel is not required but the possibility of its deformation is allowed only in its plane. A similar observation has been made in [208]. Other results of application of the Minagawa-Rado lemma and its generalizations are theorems on the second-order rigidity of troughs of revolution. The trough of revolution is determined as the surface S E C m, m > 1, produced by revolution of the curve L : z = ~v(p) about the axis Oz, where ~(p) satisfies the following properties: ~o(p) E C'n(I), m >_ 1, I : 0 < po - ~1 < p < po --b ~2;

,h,h>o,

eE1\po.

(15)

Consequently, for m > 2 the parallel L : p = p(z) is an asymptotic parallel for S. If the function ~(p) is an analytical one and ~"(po) r 0, then such a trough of revolution enters into the class of troughs investigated by Rembs and thus it possesses second-order rigidity in the class of analytical fields of infinitesimal bendings (see i.6.1,I). The statement on the second-order rigidity has been obtained in [90] for troughs of revolution under the following assumption of smoothness:

ec

= 0,

the sought-for field 2:1 of the first-order infinitesimal bending belongs to the class 6' 2 and its sought-for extension z2 into the field of second-order infinitesimal bending belongs to the class C 1. In [105], the infinitesimal bendings of troughs of revolution have been studied under the minimum requirement of C 1 smoothness of the surfaces and its bending fields. In the class of C 1 smooth troughs S with the meridian (15), ~,1 designates the subclass of those troughs which a priori are first-order C 1 rigid ones under the condition of fixation of the parabolic parallel L0 (for S E ~1 it is sufficient, for instance, that the meridian satisfy the conditions of the Minagawa-Rado lemma and its generalizations). It is not required that the trough lie on one side of the plane of the parabolic parallel L0. 1013

Among the results obtained about the infinitesimal bendings of such troughs, we shall mention the following ones:

1. In order that a trough S E ~1 n C k, 1 < k _< oo, admit the first-order infinitesimal bendings of the class C m, 1 ~_ m ~_ k, it is necessary and sui~cient that the function

g(p) =

r

-

(p0)

,

p # p0,

g(po)= 0

belong to the class C m - l ( I ) . 2. There exists a trough S E ~1 f3 C ~~ with C 1 rigidity of the first order. 3. For any m _> 2 one can be indicate a trough S E ~,1 f3 C ~ allowing C m-1 smooth infinitesimal bendings of the first-order and first-order rigidity for C '~ smooth one. 4. If a trough S E ~ l admits the first-order infinitesimal bending of the C k smoothness class, 2 < k < ~ , then the trough itself belongs to the class C k. 5. Each trough S E C 1 possesses C 1 rigidity of the second order. We call attention to properties 3 and 4. It is common knowledge that if the surface curvature K is positive, then the smoothness of the field of infinitesimal bending will coincide with the smoothness of the surface (in Hblder's classes of smoothness, see i.3.3,I), but if K = 0 is on the line, then, according to 3 the existing field of infinitesimal bending may already be made arbitrarily small compared to the smoothness of the surface. Property 4 assumes, t h a t the smoothness of the field of infinitesimal bending cannot be higher than the surface smoothness. Another investigated class of surfaces with alternating curvature is the so-called "corrugated" surfaces of revolution. They have been introduced in [105] and their meridian Z = qo(p),

O <_ a < p < b < oo

satisfies the following conditions: (1) ~(p) e C1; (2) ~'(p) = 0 in a countable number of points accumulating at a (or at b when b < or (3) [~'(p)[ has only one local minimum between two neighboring nulls. These surfaces may be doubly connected (if a > 0) as well as simply connected (if a = 0 and ~ ( p ) e C ' [ a , b]). It turns out that these surfaces are rigid relative to CLsmooth infinitesimal bendings of the first order without any boundary condition (but in the case of a = 0 it is assumed that p,Jpn+l ~ 1 as n ---* oo if p,, ~ a; in the absence of this condition there may be no rigidity (see i.2.2,I)). The boundary of the domain for the above-discussed domains containing complete parallels always consists of one or two complete parallels. The cases where curves of a certain special class are admitted as the boundary of the domain are studied in [39] a n d [40]. We shall briefly describe the conditions imposed on these curves. Let the zone S of negative curvature be on the complete surface of revolution (of sphere or torus type). The first- or second-type admissible closed line is determined by a certain rule with regard for the lattice of asymptotic lines on S. This determination is such that every curve of the first type is at the same time a curve of the second type (it is necessary to note that any complete parallel on ,~ refers to curves of the first type). Let us now consider a domain S of a piecewise class C 1 on the surface S, such that in the case of a sphere its boundary 0S has one or two components of the admissible type (see Fig. 7a), any of which is non-homotopic to null on its own zone, but in the case of a torus, the boundary 0S is an admissible one

1014

.....

T r - - :

Fig. 7a

Fig. 7b

of the first type and consists of a finite number of components arranged on one zone 5, and any of them is homotopic to null on S (see Fig. 7b). It turns out that the effect of a fixed parallel, determined in the Minagawa-Rado lemma, extends also to the domains S, in this case, any admissible line C' can be fixed instead of the parallel, but C must be nonhomotopic to null on its zone (in particular, C may be a component of vo~'); moreover, if C is an admissible one of the second type, then it is sufficient to fix only a definite part of it; if any component of the boundary 05" is free, it should be an admissible one of the first type. It is worth noting that the domains with a boundary which does not present a complete parallel are investigated for a torus also in [116]. It is proved that if a simply connected domain with a certain condition on the boundary and with K < 0 is removed from the torus, then the remaining part of the torus is rigid. In [34, 35], Ivanova-Karatopraklieva studies the first-order infinitesimal bendings of regular surfaces of revolution with alternating curvature with meridian L : p = Az '~' (z0 - z) m~ + ~(z),

~(z) _> 0,

z e [0, z0],

A is a parameter, 0 < A < ec, rnl,rn2 > O. The boundary of the surfaces under consideration consists of one parabolic parallel of second or third genus in the case of simply connected surfaces S~, and of two such parallels in the case of doubly connected surfaces S~, while the common pole of the surfaces S~ may be smooth as well as conical. If the number of first-genus parabolic parallels on these surfaces is finite and there is no any internal asymptotic parallel, then in both families of surfaces S~ and S~ there exists a countable set of first-order nonrigid ones, but all of them possess second-order rigidity. The infinitesimal bendings of doubly connected zones of a circular torus are studied in [54-56, 72, 73]. More general doubly connected surfaces of revolution of the class C 2 (including the trough of revolution) under boundary conditions of the type of sliding along the boundary were studied in [121-124, 115, 116, 208], but the results obtained in some of these works need refinement. We shall also note here the paper of N. I. Bakievich [13], where the first-order infinitesimal bendings of the troughs of revolutions of the class C a (doubly connected and noncompact ones) have been studied with the help of the characteristic equation of the field of revolution y, when the surface is glued together with a flat piece along a certain parallel, and the character of the surface rigidity is determined in dependence on the arrangement of the glued parallel. A number of papers is dedicated to the first-order infinitesimal bendings of simply connected and doubly connected surfaces of revolution of the class C 2 without asymptotic parallels. The case in which a certain 1015

parallel of such a surface is fixed relative to points, or the restrictions allowing displacement of the parallel's points only in a given direction are superimposed, is studied in [68]. The second-order rigidity of these surfaces is proved under certain restrictions on the variations of some geometrical values of the surface along their arbitrary parallel [83, 89, 37, 41] as well as under various kinematic requirements on deformations, when an arbitrary parallel is fixed relative to a point and a plane or two points, or t h e connections admitting displacement of its points only in a given direction are imposed on the parallel; see [70, 136]. If the domain on a surface of mixed curvature contains no complete parallels, another formulation of the problem should be sought. As a rule, problems of the Tricomi type are set up for the corresponding domains and certain conditions of rigidity are obtained for them [117, 121,215]. Some papers are devoted to the problems of bendings of surfaces of revolution in the classical local variant, in which the domain contains neither a pole nor a complete parallel. In [195], for surfaces of constant mean curvature, certain conditions are determined which make them isometric to surfaces of revolution. Bendings with preservation of the main curvatures are investigated in [204].

8.

BENDINGS

AND INFINITESIMAL

BENDINGS

OF POLYHEDRA

The theory of bendings of polyhedra :has its own specifity, which lies in the fact that the equations of bendings as well as of infinitesimal bendings comprise an algebraic and not a differential system. Moreover, many problems, methods, and results of their solution allow, as a rule, a visual geometric representation. We consider the polyhedra as being given a priori; thus we will not go into problems of isometric immersion of the so-called polyhedral metric into R s. As in some works, polyhedra with self-intersections are studied; in the general case, by a polyhedron in R 3 can be meant a continuous mapping of the body of a certain CW-complex of dimension 2 into R 3, which is linear on each cell, while each cell, in turn, is considered to be a closed domain with Jordan polygonal bound in a certain affine surface, and linear structures on the identified links of the boundary of two cells must be linearly isomorphic relative to the operation of identification. However, such formally complicated situations will not be encountered in the present review. 8.1.

E q u a t i o n s of B e n d i n g s a n d I n f i n i t e s i m a l B e n d i n g s o f P o l y h e d r a

When it comes to bendings or infinitesimal bendings of order n of a certain polyhedron with rigid faces it is implied that the restriction of the corresponding deformation of the polyhedron to any of its faces is the trivial bending or trivial infinitesimal bending of order n of this face, that is, the corresponding deformation of the face is either a motion or the initial Taylor expansion of a certain Cn-smooth motion of the face. By this is meant, in particular, that only dihedral angles undergo alterations in bendings, but under infinitesimal bendings of order n all points of each face should stay on a certain plane with a precision equal to o(t;~), --* 0, with the corresponding alteration of the dihedral angles. The difference between the bendings of a polyhedron as a polyhedron (that is, in the class of polyhedrons with rigid faces) and the bendings of a polyhedron as a surface can be illustrated by the example of the first-order infinitesimal bending of a flat polygon S with a fixed boundary. Let S be represented as the corresponding domain on the plane (x, y). Then S as a surface allows nontrivial infinitesimal bendings which are determined by a field of type {0, 0, ~(x, y)}, where ~(x, y) is any function of Lipschitz class C ~ given in the domain S and equal to null at its boundary. But, at the same time, S is infinitesimally rigid as a polyhedron, because during this deformation the points of the face S take the positions {x, y, t~(x, y)} which do not lie on any plane with precision equal to o(t) or, to put it otherwise, because any motion has initial speed in the form (A + [~ x r]), the field of infinitesimal bending {0, 0, ~(z, y)} does not coincide with A + [ ~ x r] for any constant vectors A and ft. The rigidity of S as a polyhedron is preserved also under the condition of its fixation only at the vertexes. There exists a very simple example of a polyhedron nonflexible as a polyhedron but flexible as a surface: it is a rectangular sheet of paper with one fixed side. As a polyhedron it allows only revolutions about the fixed side and as a surface it can be rolled up in a cylinder with preservation of the fixed side on its place. 1016

Another simple but more interesting example is the bending of a tetrahedron proceeding in the following way: one of the faces of a tetrahedron T is located on the plane (z, y) and its opposite vertex is located on the plane z = h > 0. Let us draw a plane lit : z = h - t, t > 0, and mirror the upper part of the tetrahedron intercepted by the plane with respect to the plane IIt. In the case of the variable t, 0 < t < h, a continuous family of polyhedrons Tt with seven vertexes, which are isometric to the original tetrahedron T, is obtained. This bending takes place in the family of polyhedrons of one and the same combinatorial structure, but in this case some faces are deformed by bending along the variable line of reflection. That is why such a bending also does not refer to the deformations which are usually investigated in the theory of bendings of polyhedrons (for a more complex example of a similar bending see, i.8.4). The infinitesimal headings corresponding to this bending will be discontinuous on the variable line of reflection. Such a situation is met in the so-called supercritical deformations of membranes and, therefore, the investigation of such bendings is of certain interest (see, for example, [93]). In addition, a proposition has been made in [212] that all isometrical deformations of a polyhedron are exhausted by such mirror images. It is common knowledge now that this proposition is a wrong one, but undoubtedly there is something deserving of attention in this proposition. The special character of admissible deformations in the bending of a polyhedron in the class of polyhedrons leads, in the general case, to certain difficulties in the analytical description of such deformations. The fact is that a polyhedron in a space is completely determined by the coordinates of the vertexes and its given cellular structure, that is, by the combinatorial scheme of the combination of vertexes, edges, and faces. But if the polyhedron undergoes deformations in the class of polyhedrons, the alterations of the coordinates of its vertexes should be correlated so that the vertexes of one face stay precisely or with the required accuracy on one plane. This fact should be reflected in the corresponding equations. But if the faces are triangular the last problem is eliminated: three points always lie on one plane. In line with this assumption, the equations of bendings and infinitesimal bendings of a polyhedron are usually written under the supposition that all faces are triangular. They have the following from:

Ivy(t)- pAt)l

(

t"z

p~

-

= Ip~ -p~

tmz~

+2

rn=l

(i,j) E/~ =[p~176

(16)

).

(17)

m=l

0 Here pO = (xi,0 Yi, z~ are the points of the polyhedron under study with V vertexes, pi(t) with pi(O) = pO are the positions of the vertexes during deformation, depending on the parameter (of time) t; the inclusion

(i,j) E /~ signifies that Eqs. (16) and (17) are written only for pairs of vertexes connected by edges (the number of edges will be denoted by E); z~i, 1 < i < V, 1 < m < n, are vectors determined respectively in the vertex p0, and giving the infinitesimal bending of order n as a deformation of the following type: v,(t) = v ~ + 2

t

t

o.

(18)

rn=l

Notice that now, if one wants to describe the corresponding deformation of the polyhedron as of a surface with a given p,(t), satisfying Eqs. (16), (17), and (18), it is done unambiguously with regard for the requirement of the rigidity of the face: a point (z ~ yO, z o) Of the face with vertexes pO, pO, pO, which has barycentric coordinates Ai, Aj, Ak,~transforms into the point (x(t), y(t), z(t)) = Aipi(t) + Aipj(t ) + Akp~(t). But without the condition of the rigidity of the face it is evidently impossible to regain the required deformation of the surface. The restoration of the deformation may appear to be impossible also in the case of a rigid but not triangular face. It is necessary to note that the requirement for faces to be triangular is not a very strong restriction: the condition of the rigidity of a non-triangular face implies that the diagonals inside the face should preserve their length (with the proper accuracy) and, therefore, the non-triangular face can be broken down by its 1017

diagonals into triangular ones. If a polyhedron P triangulated in the above-mentioned way is an infinitesimally rigid (inflexible) one, then the initial polyhedron Po will be all the more infinitesimally rigid (inflexible). But the inverse is incorrect. Thus, there is no comprehensive information on P0 in the case of the infinitesimal nonrigidity (flexibility) of P. However, in principle, the equations of the infinitesimal bendings for polyhedrons with arbitrary faces can also be written. For this purpose it is necessary to recall the general representation of one-parametric motion in the form of Taylor's series (see, for example, i.3.1 in [113]), and then to require the matching of constant vectors participating in this representation for each face in such a way that all faces convergent to some vertex could provide one and the same motion for this vertex. For instance, the first-order infinitesimal bendings z of a polyhedron with F faces is determined by 2F constant vectors Ai, f~i, 1 < i _< F, in the following way: r(u, v; t) = r(u, v) + tz(u, v), z(u, v) = Ai + [f~i x r(u, v)], (19) where i is the number of the face to which r(u, v) belongs. The condition of continuity of z over the whole polyhedron provides the corresponding conditions on A and f~i on the edges and in the vertexes of the polyhedron. The equations of bendings and infinitesimal bendings could be presented in a better way than in (16) and (17). Thus, if for a polyhedron P C R3(Zl, x2, x~) with V vertexes and triangular faces a certain number of vertexes is fixed, then the point M 9 R 3v with coordinates Xll, X21, X31; * . . ; XlV, Z2V,

X3V)

can be associated with this polyhedron, and inversely, for each point from R 3v there may be found a corresponding polyhedron of a given combinatorial structure in R 3. According to this association, a certain path M(t) of point M in R 3w corresponds to any continuous deformation of the polyhedron P in R 3, in particular, to its bending. In this sense, for a polyhedron, its bending as a whole appears as an object of finite-dimensional geometry, and the terms "C"-smooth-by-parameter bending," "analytical-by-parameter bending," and so forth, assume their habitual meaning as "Ca-smooth path M(t) in R3V, " "analytical path in R3V, " and so on. In such an interpretation, the assignment of the lengths dij of the polyhedron edges connecting the vertexes with the numbers i and j are written as the following system of equations: (Xli

-

Xlj) 2 .JI- (x21 - x2j) 2 + ( x 3 / -

x3j) 2

d ,,j, .2 .

(i,J) 9 E.

(20)

Consequently, it immediately turns out that the set of polyhedrons isometric to a given polyhedron P0 (a complete configurational space of the polyhedron) represents an algebraic surface in R 3v, and the set of positions assumed by the polyhedron P0 in the process of its bendings (the connected configurational space To of the polyhedron) is described as a connected component of the algebraic surface (20) containing the polyhedron Po. Except for nontrivial bendings, the motions of P0 as a solid body also appear in the space To. Thus, in order for To to contain only nontrivial bendings, six additional conditions excluding the motions of Po must be imposed on the deformations of Po. The first three equations E Xli = 0, i

E

X2i : 0,

E 2:3i "~" 0 i

/

(21)

fix the position of the center of mass at the origin of the coordinates, but the problem of excluding the revolution around the mass center "bifurcates." If the polyhedron P0 with conditions (20), (21) is recognized as a given one, then, according to [186], other polyhedrons with vertexes yi, isometric to the polyhedron Po, can be searched for with conditions of the same type (20), (21) supplemented by the equation x i

1018

=o.

(,)

If the problem of nontrivial bending is considered simultaneously with the problem of isometrical immersion, then the Eq. (20) with the given dlj, (21), and three more equations = o, i

F Xl,X3, = o, i

E i

= 0

(22)

can be solved. Then no two solutions of the system (20)-(22) from one connected component will be connected by continuous motion, excluding the cases where the ellipsoid of inertia of the set of the polyhedron vertexes is the ellipsoid of revolution, specifically, the sphere. Two points need to be made in relation to the system (20)-(22). R e m a r k 1. If dij are taken as variable ones with normalization ~ ~i = 1 (for exclusion of similarity), we obtain from (20) that the set of polyhedral metrics on a given simplicial complex, isometrically immersable into R 3, represents a certain semialgebraic variety with boundary (the projection of the variety (20) from R 3v+E iuto RE). It seems likely that the nature of this variety M0 in a more or less general case has not been studied yet. Some data on the variety M C M0 corresponding to the metrics of positive curvature can be found in [3]. R e m a r k 2. It is of interest to study the problem of the isometric immersion of a convex polyhedron metric as a solution of the algebraic system (20), that is, to determine the effect of the convexity on the solvability of the system (20). To put it in another way, the geometrical language of the reasonings of induction by the number of vertexes, which has been applied in [3], should be translated into the language of algebra. The equations of infinitesimal bendings of polyhedrons (with triangular faces) allow two representations a kinematic (in terms of motions and speeds) and a static one (in terms of force and tensions). In the first representation, the field of infinitesimal bending is searched for as a set of vectors zi, applied in vertexes xl and satisfying the system of equations

-

-

(z, -7 zj)(xi - xj) = O,

(i,j) e k

(23)

(the trivial infinitesimal bendings are excluded by the conditions Ezl = 0

and

~'~[xi • zi] = O, i

obtained from (21)-(*)); the field zi determines the infinitesimal bending as a deformation of the following type: xi(t) = x, + tz, + o(t), t ~ O. (24) Before proceeding to the second representation of the system of equations of the infinitesimal bending, it is necessary to note that the field of the initial speeds zl of infinitesimal bending (24) determines the speed of alteration 5~j of the dihedral angles at each edge { i , j ) . In this case, 5~ij are connected by the equalities E~3ij (xi-xj)

where the summation is performed over all edges { i , j } going out of the vertex zi. With an obvious change in designations, this system can be represented in the following form: ~,j(xl-xj)=O, J

1 < i < V.

(25)

Exactly the same system describes the case where the forces wlj(xi - x j) with wlj = wjl are applied along each edge going out from xi, provided that these forces are in equilibrium at every vertex xi. For example, such forces appear in a balanced system of elastic bars under internal stresses. In view of this analogy, the 1019

set of real numbers wij = wji defined on the set of edges is called the stress (of the polyhedron framework); moreover, an edge is said to be under strain or under compression if wij < 0 or wij > 0 respectively. If the system (25) is valid, that is, the system of internal forces is in equilibrum, the real numbers wq are called self-stresses. A system of external forces Fi applied in the points xi is said to be balanced if E Fi = 0 and ~[Fi x zi] = 0. The stresses wij on the edges of a polyhedron framework are said to solve the balanced system of forces Fi if Fi + ~ wq(xi - xj) = O, J

(26)

1 < i < V.

Further, the framework is called statically rigid if it solves any balanced loading Fi. From the viewpoint of mechanics, this means that any external balanced load causes an internal distribution of stresses along the edges without any alteration of the framework shape. If the framework is not statically rigid, it cannot statically balance the external loading by means of internal stresses; therefore the framework has to react dynamically, that is, by changing its shape. The relation between static rigidity and kinematic (or ordinary) infinitesimal rigidity is established by the following remarkable theorem. T h e o r e m . A framework is rigid with respect to first-order infinitesimal bendings if and only if it is staticaI1y rigid. This theorem is valid not only for the frameworks formed by the edges of polyhedrons, but also for any framework made of bars in the Euclidean space of an arbitrary dimension; for a proof, see [164, 203]. The validity of this theorem for polyhedronw follows from the fact that the rigidity of a polyhedron with triangular faces is equivalent to the rigidity o f the framework made up of its edges. Thus, the second representation of the equations of infinitesimal headings requires the investigation of solvability of the system (26) under any balanced system of forces Fi. However, if nonrigidity is established in this way, the determination of the field of infinitesimal bendings requires additional considerations. The determination of the self-stress wli from (25) through a given field zi of the infinitesimal bendings is rather simple (through the variations of dihedral angles), but the inverse procedure is not obvious, and cannot be realized in the general case. The formal reasoning is as follows: the equation s - m = e - vd is valid [164] for the dimension s of the space of solutions of the homogeneous system (25) and the dimension m of the space of solutions of the homogeneous system (23); here e is the number of edges of the framework, v is the number of vertexes, and d is the dimension of the space R d where the framework is essentially contained, that is, it is not contained in any R d-1 C R d. Consequently, if the framework consists of the edges of a polyhedron of the genus p _> 1 and the number of its linearly independent nontrivial infinitesimal bendings is equal to m0, then s = 6p + too, i.e., if 0 < s _< 6p, then m0 = 0 and the polyhedron is rigid. The analogy with the determination of a field of infinitesimal bendings z through the rotation field y in the theory of bendings of smooth surfaces is the informal explanation of the aforesaid: it is shown in [27] that additional conditions should be superimposed on the field y in the multiply connected case, guaranteeing the single-valuedness of the field z defined through the differential form dz = [!1 x dz]. Similar conditions for frameworks have been considered in [171], but the problem of revealing the minimal number and type of such conditions remains open. In the case of a sphere, s = m0 ([176]), but even then the practical determination of zi through wij is a nontrivial problem in view of the dependence 'of the algorithm on the polyhedron's combinatorial structure. The interpretation of a polyhedron with triangular faces or more generally that of a framework as a point in R TM (d is the dimension of the space where the framework is situated, and V is the number of vertexes of the framework) allows one to consider the metric of the polyhedron as a certain mapping from R TM into R E, where E is the number of edges of its framework. This mapping f : R TM ---* R E is called the rigidity map and is defined as follows (see [142, 161,203]):

y: (...,p,,...,pj,...)

--, (...,(p, - pA2,...),

(i,j) e

This mapping is highly informative for a general description of the set of infinitesimally rigid Or inflexible 1020

polyhedrons and frameworks. This fact may be attributed mainly to the following: the mapping L p : R 3v ---, R e ,

L p ( ~ l , . , . , ~v) = ( . . . , (p, - pj)(6i - ~ j ) , . . . ) ,

considered initially in [176] for polyhedrons in R 3 and then extended to the general case R d, d > 2, is the derivative of the mapping f, and the system of equations of infinitesimal bendings (23) defines the kernel of the mapping Lp. Moreover, the system (25) may be connected with the transposed matrix of the mapping Lp or with the operator conjugated with Lp. Therefore, application of the known theorems of linear algebra and the implicit function theory makes it possible to obtain a certain classification (by rank of the matrix of the mapping Lp = ldfp) of the points in R dv which are considered as polyhedrons with lengths of edges (pi - p j ) 2 = ~j, and to connect the infinitesimal rigidity and inflexibility. For example, it is proved for frameworks in [142, 143] that: a framework in R d is infinitesimally rigid r rank dfp = V d - ( m + 1)(2d - m)/2; a framework in n d is infinitesimally nonrigid ~:~ rank dfp < V d - ( m + 1)(2d - m)/2, where rn < d is the dimension of the subspace R m C R d where the framework is actually contained (in an exact sense). Further, a framework is infinitesimally rigid in R d <=~ a framework is inflexible in R d and the corresponding point p E R TM is a regular one of the mapping fp, in which the derivative dfp has the maximal rank, that is to say, the properties of infinitesimal rigidity and inflexibility coincide at the regular points. 8.2.

Cauchy's T h e o r e m

and Its Generalizations

Survey articles on the bendings of polyhedrons traditionally begins with the exposition of Cauchy's Theorem about the unique determination of a closed convex polyhedron in the class of convex polyhedrons, accompanied by the more or less latest commentaries. W e hold to this tradition and also begin the review with this theorem. The work [175] is devoted to the history of Cauchy's Theorem. Taking into account that in [175] and many other works Cauchy's Theorem is associated with the contents of Definition 10 of the XIth b0ok of Euclid's "Elements" on the conditions of the congruence of polyhedrons (as bodies), we mention also the article [140], where, among other things, a not purely mathematical meaning of the enumeration of all regular polyhedrons as the main result of Euclid's "Elements" is justified. Using the modern terminology and understanding of the relationships between physics (in a broad sense) and mathematics, and strongly simplifying the contents of Plato's explanation of the universe, the idea of [140] can be formulated as follows: Euclid gave the mathematical bases for the theoretical views of Plato, who described matter as various combinations of basic elements - - fire, air, earth, and water - - which have the geometrical forms of tetrahedron, octahedron, cube, and icosahedron respectively (the dodecahedron was associated by Plato with the earth visible from the cosmos). Since, according to Euclid, there are no other regular polyhedrons except the five enumerated ones, this fact is the mathematical reasoning for the universal character of the basic elements corresponding to the regular polyhedra. As we see, there was a period when the problem of the unique determination of polyhedrons was of fundamental scientific significance. In the thorough investigation of Euclid's "Elements" [180], one can find a detailed analysis of the classical works on the interpretation of Definition 10 from the XIth book, including the works by Simson and Legendre, who consider the contents of this definition to be a theorem. The analysis of the relations between Cauchy's Theorem and the Euclidean definition of the equality of polyhedrons is also presented in [60], in which the author justifies his point of view that Euclid perceived the equality of polyhedrons (as bodies) as the equality of their volumes. Cauchy's Theorem and its proof were included in a number of textbooks on geometry (see [147], for example). 1021

Further, we pursue the established tradition to report on the mistakes of fatally straying from Cauchy's Theorem. A discussion of Cauchy's Lemma on two convex isometric polyhedrons and critical notes concerning its proofs can be found in [205]. A number of critical notes on various proofs and interpretations of Cauchy's Theorem could have been presented b y the authors of [178], but unfortunately [178] has the form of a concise annotation. Certain references testify that the publication [177] also contains a critical discussion of questions concerning Canchy's Theorem, but this work was not available to us. The absence of attention to the exact meaning of the notion of a convex polyhedron is one of the common negligences in presentations of Cauchy's Theorem and its infinitesimal analogies. It is often not clear if the convex polyhedron is considered to be a strictly convex one or if it may have "false" edges (with dihedral angles equal to 7r) or "false" vertexes (with polyhedral cones degenerated into a dihedral angle with the vertex on the "real" edge or even into a plane). Ignoring the special role of "false" vertexes may result in wrong conclusions. For example, the following mistake in the proof of Lemma 5.3 in [176] concerning the case (b) was detected by B. Grunbaum: actually, even a "real" vertex may have index 2 if "false" edges emanate from it (see [3], Ch. X, w L e m m a la). In our survey, by a convex polyhedron is meant a strictly convex one. The incorrectness of the proof of Steinitz' Theorem in [52] on the realization of any triangulation (or even cell decomposition with polyhedral cells) of a sphere by a convex polyhedron in R 3 is investigated in [146], where another proof of this theorem is offered (the proof of Cauchy's Theorem has no need of Steinitz' Theorem, but the last one is widely used in a number of other theorems on the bendings of polyhedrons). It is well known that Cauchy's Lemma on two isometric convex polygonal lines (on a plane or on a sphere) is of considerable importance for proving Cauchy's Theorem. It even has a special name - - the Opening Arm Lemma. Many authors have offered new proofs of this lemma (see [60, 173, 205, 211]). The Schur's generalization of the lemma for smooth curves, one of which is plane and convex, and another of which may be evenly arranged in the space, is considered in [153]. In this work, a variant of the same lemma for piecewise regular curves is also formulated. We recall that in [148, 150] a smooth analog of the following corollary from Cauchy's Lemma had been presented: the difference of dihedral angles on two isometric convex polyhedral angles changes its sign at least four times when going around the vertex. A concise survey of the works on Cauchy's Lemma can be found in [167]. In the same work, one more proof of the lemma based on the uniqueness of the so-called Cauchy polygon is also presented with reference to [162]. Since the uniqueness of the Cauchy polygon holds even in the space R '~ of any dimention n > 2, it turns out that Cauchy's Lemma is valid for a plane convex polygonal line and for a polygonal line isometric to it, a priori contained in any R '~, n>2. We now proceed to a closer consideration of Cauchy's Theorem itself. Besides the modifications of the original Cauchy proof of the theorem and the extension of this theorem to some cases of nonconvex polyhedrons (see, for example, [186, 211]), there have appeared principally new proofs of the theorem. A n e w proof from [4] is based on the theorem on the uniqueness of a general nonclosed convex polyhedron P provided that its boundary is fixed with respect to some point O from which the whole interior of P is visible, and which can be separated from P by a plane passing between them. Here "fixing of the boundary" means that the distance from O to the points of the boundary of P is preserved under an assumed isometric deformation. The proof of Cauchy's Theorem in [114] is also based on the theorem on the uniqueness of a general convex polyhedron with a boundary fixed relative to a point, but the proof of the theorem is carried out in another way, with application of the condition of equality of the convex polygons on the sphere established in [28]. A particularly short proof is given in [92]. This proof is based on the following lemma from [24]: if two strictly convex polyhedral angles P and P ' are isometric, Aal = a~ - ai is a variation of the dihedral angles with the edges numbered i, and e~ is the unit vector of the ith edge originating from the vertex of the polyhedral cone, then the vector ~ Aalei is directed toward the interior of the spherical image of the angle P. Besides the problem of the unique determination, there exists the question on bending of convex polyhedrons. Cauchy's Theorem positively solves the question on nonflexibility of the closed convex polyhedrons only in the case of their strict convexity. This is also valid if a polyhedron is not strictly convex, but continuous 1022

":"

~, ;'

,

Fig. 8 isometric deformations take place in the class of convex polyhedrons. However, the bendings destroying the global convexity are a priori possible in the case of "false" vertexes or edges in the polyhedron. Therefore, in the general class of bendings, even Pogorelov's Theorem on the unique determination of general convex surfaces cannot solve the problem. However, it turns out in the case of polyhedrons that if a polyhedron is flexible, then it is also flexible in the class of analytical-by-parameter deformations. By using this property, the inflexibility of a closed and convex polyhedron obtained from a strictly convex polyhedron under any of its triangulations is shown in [161]. It is proved in this work that if a convex polyhedron does not contain "false" vertexes, then it possesses first-order rigidity ([3]), and if it has "false" vertexes, then the first-order rigidity fails, but the second-order rigidity is valid. These facts imply the analytical-by-parameter inflexibility (see i.3.3,I). Another proof of the inflexibility of a general polyhedron in the class of analytical-by-parameter deformations is presented in i.3.3,I; this proof does not assume that the faces are triangular and the deformations are realized in the class of polyhedrons, that is to say, the rigidity of the faces is not assumed here. But the general problem on the inflexibility of strictly and nonstrictly convex polyhedrons with nonrigid faces in the class of deformations with non-analytical dependence on the parameter still remains unsolved. The proof in Item 3.3,I is simple, but it is based on the very complicated A. V. Pogorelov Theorem about the rigidity of an arbitrary convex surface. The correlation between this theorem and the results from framework theory is of particular interest: a framework (a set of edges) of a convex polyhedron (without vertexes with plane polyhedral angles) is infinitesimally rigid if and only if the faces of the polyhedron are triangles. In particular, the framework of a cube is infinitesimally nonrigid. But the same framework considered as a part of a polyhedron is infinitesimally rigid by Pogorelov's Theorem. 2 Moreover, it is infinitesimally rigid provided that it is considered a~s a part of a nonclosed polyhedron constructed from the cube in the following way: on every face of the cube an open domain is cut out whose closure does not contain the interior points of edges (see [3, 16]). Thus, if the cut-out domains of the faces are squares or rectangles, then P may be considered as a framework composed of profiles in the form of angles rather than bars (Fig. 8). Note that in the general case the rigidity of such "framework strips" has not yet been studied. This fact is also remarked in [202] as a consequence of the view of the infinitesimal bending of a polyhedron with faces rigid in their planes (see the end of the present section).

1023

The proof in [161] is rich in technical innovations. It gave rise to many works devoted to the rigidity of frameworks. A fruitful idea on the relation between the rigidity of a polyhedron and the rigidity of each of its faces in the plane containing the face has been used in this work. Other results obtained in this llne of investigations are presented in [203]. All or almost all of the above-mentioned proofs of Cauchy's Theorem are appropriate for the establishment of the first-order rigidity of a closed convex polyhedron. A number of works are particularly devoted to the infinitesimal variant of Cauchy's Theorem - - the Den's Theorem. An analogue of the Blaschke integral formula for a closed polyhedron or a polyhedron with a border is offered in [58]. It is based on formula (19) and can be applied for verifying the infinitesimal rigidity of a closed convex polyhedron. A new approach to the problem of the infinitesimal rigidity of a polyhedron is suggested in [126] and [127]. If for every vertex pi there exists a speed vector vl, then the average point q of the edge plpj is singled out and the half-edge piq is painted dark blue, red, or green, depending on whether the scalar product plq and vi is positive, zero, or negative respectively; thus, all half-edges turn out to be painted in one of the three colors. In [126] and [127] a number of theorems on the admissible combinations of colors of the half-edges for the orientable polyhedrons of any topological type is established. Then, on the basis of these theorems, it is stated that if a convex polyhedron is deformed under the effect of the field of its infinitesimal bendings, all of its half-edges must be painted red, and in this case all vi = 0 and the field of speeds is trivial. Under certain requirements imposed on the type of the field, the rigidity of some nonconvex polyhedrons, not necessarily homeomorphic to the sphere, can be obtained. A similar approach to using other terms has been suggested just recently in [206], but we know this work only by its exposition in [167]. A number of other proofs of the Den's Theorem are also mentioned in [167]. The classical variants of the proofs of the infinitesimal rigidity of a convex polyhedron according to A. D. Alexandrov [3] are presented in [176] and [186]. The most advanced generalizations of Cauchy's Theorem on the unique determination of polyhedrons are given in [271]. For more detailed information about this work, see Sec. 8.5. The infinitesimal rigidity of a convex polyhedron under the most general conditions is established in [227] (see also [202]). In these works the frameworks made up of the "real" edges of a convex polyhedron with added edges which need not be disjoint are considered. In these cases various conditions of the infinitesimal rigidity of such frameworks are presented. Moreover, the above-mentioned interesting problem of the relations between the infinitesimal rigidity of a polyhedron in space and the infinitesimal rigidity of some faces in their planes are investigated. For example, it is shown that the infinitesimal bending of a polyhedron consists only of the vectors directed perpendicular to the faces if the faces are infinitesimally rigid in their planes. To summarize, we indicate that in the '40s and '50s Cauchy's Theorem was proved comprehensively for the unique determination of convex polyhedrons, and in the '80s the main progress was made in the complete solution of the problem of bendings and infinitesimal bendings of convex polyhedrons. 8.3.

Relations between Bendings and Infinitesimal Bendings. General Questions of the Theory of B e n d i n g s

In the case of a polyhedron, the algebraic nature of the equations describing both bendings (20)-(22) and infinitesimal bendings of the first and higher orders leads to certain remarkable relations between bendings and infinitesimal bendings which are absent in the case of general surfaces. (1) The algebraic manifold A defined by the system (20)-(22) with fixed dli > 0 has a finite number of connected components arranged in a bounded ball of the space R 3v. Therefore, each of its connected components is compact and there exists a minimal distance d > 0 between them. This implies that for any polyhedron P there exists a positive number e(P) > 0, e(P) < d ( P ) , such that if a polyhedron Q isometric to P is contained in the e-neighborhood of the polyhedron P in R a, then both polyhedrons belong to one connected component, that is, P can be nontrivially bent to Q. Thus, if P is an inflexible one, then there are no nontrivial polyhedrons isometric to it in a certain neighborhood of it; in other words, an inflexible polyhedron is uniquely determined in a certain neighborhood of it. This situation does not correspond to the 1024

general case since there exists an inflexible convex (nonclosed) surface which is the limit one for the surfaces isometric to it (see i.3.2,I). (2) If some polyhedron P is bent (in the class of continuous deformations), then it is also bent in the class of analytical-by-parameter deformations. In particular, this means that a flexible polyhedron is always infinitesimally nonrigid. The following is also true: if there is a continuous bending of P on Q, then there also exists a piecewise analytical-by-parameter bending P on Q. Apparently, attention to these sufficiently simple and almost obvious asseitions was noted the first time in [176]. The problem on the description of the manifold A for a given polyhedron was also stated there. This interesting but complicated problem is still poorly known; one can only find isolated results in [22, 108, 113]. In particular, the problem of finding (if only by an algorithm) the real numbers d(P) and r defined in (1) has not yet been posed anywhere. Note that d(P) depends only on the internal metric of the polyhedron P, and r depends on its external structure. (3) In contrast to the surfaces of revolution, the set of the infinitesimally nonrigid polyhedrons is closed. Indeed, certain determinants are equal to zero for any nonrigid polyhedron, that is, the set of nonrigid polyhedrons is an algebraic surface in R 3V. Another difference between the bendings of polyhedrons and the bendings of "curved" surfaces is in the more explicit dependence of the flexibility on the infinitesimal nonrigidity. So, in [161] with reference to M. Gromov, it is indicated that every inflexible polyhedron has infinitesimal rigidity of some order n > 1. This statement may be strengthened as follows: for any k > 1 an inflexible polyhedron possesses infinitesimal rigidity of some order (k,n(k)). One of the proofs can be obtained by using the main result from [141] residing in the fact that if a system of algebraic equations admits a polynomial approximate solution of a sufficiently high degree, then there exists an exact solution represented as a convergent series (for more detailed information, see [113]). It follows from the formulation of the theorem of [141] that for every polyhedron there exists a finite algorithm which verifies its flexibility. To verify the flexibility of a polyhedron one has to verify its infinitesimal nonrigidity for a sufficiently high order N; this procedure reduces to the consecutive solving of some linear system with the right part depending on previous approximations. But we do not know of any algorithm for determining the required degree of the approximation N. The existence of a finite algorithm for checking the flexibility of a polyhedron (and of all frameworks in R a, d > 3) based on the general theory of solvability is also advocated in [183]. The idea of an algorithm for pointing out a flexible polyhedron among all nonrigid polyhedrons is presented in [113]. The question on algorithms for checking the flexibility of polyhedrons is also discussed in [160]. An algorithm for determining the flexibility of a framework on a plane is proposed in [191] (see also [172]). An algorithm for verifying the flexibility of a suspension is described in [110], but the question about finding a general and effective algorithm for checking the flexibility of a polyhedron remains open. 3 A great number of works is devoted to new proofs of the projective invariance of the rigidity of polyhedrons and frameworks as well as to application of this property for obtaining new facts about the infinitesimal bendings of a polyhedron (see [184, 203, 219, 220, 231,239]). In [170] the infinitesimal bendings and stresses in a balanced system of frameworks and bars are studied as the objects of projective geometry. The Cayley algebra is used as the main tool of investigation, and, in particular, the projective invariance of static properties of the frameworks is established. Similar questions concerning the polar transformations of the frameworks are considered in [231']. 9 New proofs and interpretations of the Maxwell-Cremona Theorem on the determination of self-stresses in plane graphs are suggested in [167, 170, 226]. Maxwell's Theorem asserts that the projection of a sphere-type polyhedron onto a plane results in a dependent framework made up of the projections of edges, that is, the edges of the framework admit nonzero balanced self-stresses. A simple explanation for this theorem is as follows: if the structure of the faces of a polyhedron P is transferred to its projection onto the plane, then Now such an algorithm is known at least for the polyhedrons admitting only a one-dimensional space of nontrivial infinitesimal bendings; see the article by I. Kh. Sabitov in Vestn. Mosk. Univ., No. 1 (1994).

1025

the projection may be considered as a polyhedron with self-coverings. Such a plane polyhedron Po admits a field of infinitesimal bendings perpendicular to the plane of the polyhedron P0. The infinitesimal motion of any face will be its rotation around a certain straight line, provided that the values of this field at the vertexes of P0 are chosen so that the ends of the vectors zi coming out from the vertexes of P0 end up in the space in the corresponding vertexes of the polyhedron P. Then wij may be calculated from the motions (as shown in i.8.1), and treated as stresses. In this reasoning, the topological type of the polyhedron is of no importance. More refined arguments produce the inversion of Maxwell's Theorem: by t h e presence of balanced self-stresses with certain additional conditions it is possible to determine whether the framework on a plane is the projection of a framework made up of the edges of a certain polyhedron P [226]; moreover, it has been established in [226] that the type of stress determines the type of connection between the corresponding faces (convex or nonconvex) on the edge (extension or compression). This fact turns out to be important in the theory of pattern recognition. Information on the application of Maxwell's Theorem and its inversion (for example, in ~he theory of electric nets) can be found in [167] and [226]. Extensions of Maxwell's Theorem to the n-dimensional case exist, but for its inversion sufficiently general conditions for the recognition of the projection of the (d + 1)-dimensional polyhedron in R a, d > 3 have not been obtained; see [226] (here the author must say that the information familiar to him on this problem has a rather fragmentary character, since the majority of recent results on generalizations and inversions of Maxwell's Theorem are published in the form of preprints or in user-oriented editions not mentioned in reviewing journals). An analog of the inversion of Maxwell's Theorem in the one-dimensional case has been studied in [232]. Here, a set of intersecting straight lines is given on a plane, and it needs to be ascertained if nonintersecting straight lines whose projection is the given set exist in the space. The answer to this question is formulated through the rigidity of a certain system of polar-dual objects. Broadly speaking, sometimes the rigidity manifests itself as a characteristic condition in the most unexpected problems. Various applications of this notion are shown in [165, i66, 167, 233, 242]. Finally, the concluding result of the long-standing problem of the distribution of infinitesimally nonrigid and flexible surfaces among all surfaces has been obtained. It is shown in [96] that almost all sphere-type polyhedrons are infinitesimally rigid (the idea of its proof is the following: the matrix of the system (23) has maximal rank almost everywhere; the points where the rank is less than the maximal one form a certain surface in RZV). It is indicated in [176] that the infinitesimal rigidity of polyhedrons implies their inflexibility (for a detailed explanation, see Sec. 8.2); therefore, the set of flexible surfaces is "less" than the set of infinitesimally nonrigid ones, and the first set forms a certain algebraic submanifold in R 3v. The extension of the result of [96] and [176] to the surfaces of torus type and more generally to the surface of high topological genus turned out to be a complicated problem. This problem was solved in the late 1980s: first it was shown in [229] that almost all of the polyhedral tori were rigid, and the general solution was obtained in [174]. The idea proceeds [230]: if a polyhedron P with V vertexes is rigid in the general position, then the polyhedron P1 with V + 1 vertexes obtained from P bj, splitting one vertex is also rigid in the general position. Thus, it remains to show that any polyhedron of a given topological type can be obtained by consecutive splitting of the vertexes of some minimal irreducible polyhedrons whose rigidity may be established by special consideration. Apparently, this remarkable theorem is still waiting for a more simple proof. 8.4.

Flexible a n d N o n r i g i d P o l y h e d r o n s

Although the question on the existence of flexible embedded or, at least, immersed closed polyhedrons had been being under discussion almost for two centuries, an example of such a polyhedron was found only recently. At first, R. Connelly [157] constructed an example of a flexible immersed polyhedron with a selfcontact transforming into the self-intersection under bending. Later, he devised a method for removing this self-intersection, and thereby obtained an example of a flexible embedded polyhedron in 1978. The description of the flexible embedded polyhedron and its modifications can be found in [29, 53, 158, 159, 186]. 1026

A particularly simple example with thus far an extremely minimal number of vertexes equal to 9 is proposed in [21014; this polyhedron is also described in'many other works: [113, 147, 159]. In order to gain a comprehensive notion of the possible variety of flexible polyhedrons, the following points should be investigated: the minimal number of vertexes of the embedded (or immersed) flexible polyhedron of an arbitary topological type; the maximal realizable dimension of the configuration spaces as a function of the number of vertexes and edges; the structure of the configuration spaces in the neighborhoods of the singular points of the algebraic surfaces which represent these configuration spaces. A theorem of a very general type describing the set of all flexible polyhedrons (more exactly, frameworks) with a given number of vertexes as a topological space is given in [183]. The sums of the Betti numbers of this space are estimated from above as a function of the number of vertexes. In particular, the upper bound estimate of the number of connected components is given, but there is no concrete information on the structure of the configuration spaces. In [113], the configuration spaces of the plane quadrangles are described; a description of the configuration spaces of the flexible Bricard's octahedrons is given in [22].s The following comparison between the process of cognition and the extending sphere is well known: all of what is in the interior of the sphere presents the knowledge obtained, but we are not even able to raise the question of what is in the exterior of the sphere, and problems are formulated only on the boundary between knowledge and ignorance. The history of the discovery of the existence of flexible polyhedrons confirms the validity of this comparison. Only the construction of the examples of the flexible polyhedrons disclosed their remarkable property: the inside volume of the polyhedron remains unchanged in the course of bending. This property is an inherent characteristic of all known flexible polyhedrons. This fact naturally resulted in the Connelly-Sullivan problem where it must be proved that any closed flexible polyhedron does not change its generalized volume in the course of bending [160]. The generalized volume of the embedded polyhedrons coincides with the usual volume,~ while the generalized volume of the immersed polyhedrons appears to be a formal application of the volume formula as the sum of the algebraic volumes of pyramids with a vertex fixed in a certain point and a base lying on the faces of the polyhedron. This hypothesis has not yet been proved. One of the possible approaches to it would be the proof of the assumption that the generalized volume of a closed polyhedron is the root of a certain polynomial with coefficients depending only on the lengths of the polyhedron edges. If this assumption is valid, it would be possible to determine the values of the volume of the future polyhedron already before constructing the isometric immersion of the given polyhedral metric in R 3. There exist certain strong grounds for believing that such an assumption is valid. Another approach to the Connelly-Sullivan hypothesis lies in studying the behavior of the volume of a polyhedron under its infinitesimal bending. If the volume of the polyhedron were stable under its infinitesimal bendings, the hypothesis would have been proved. However, it has been shown in [6] that there exist infinitesimally nonrigid polyhedrons with unstable volume, and this volume varies over a precise order O(t), t ~ 0. Moreover, examples of polyhedrons with second-order nonrigidity can be shown, whose volume is also unstable. Since the polyhedrons are inflexible in all these examples, it is necessary, at first, to detect the infinitesimal bendings afortiori extendible into the bendings, and then to verify their volume stability. But this way seems to be a complicated one. The set of infinitesimally nonrigid polyhedrons is more rich than that of flexible ones, and consequently we have more examples of infinitesimally nonrigid polyhedrons. A large number of infinitesimally nonrigid polyhedrons has been studied in [5, 234, 238, 240, 243, 245]. The class of toroidal polyhedrons is studied in [145]. These polyhedrons are constructed in the following way: a given convex closed polygonal line on a half-plane xOz, x > 0, is juxtaposed to its three other positions produced by rotation of the line about the axis Oz through the angles ~, 7r, ~" from its starting position, and then the corresponding vertexes are connected by the intercept of straight lines. For such polyhedrons 4 I. G. Maksimov has now proved that this number 9 is minimal for the existence of an embedded flexible polyhedron.

s The works of T. F. Havel and B. Jaggi are also devoted to this subject.

1027

A3

A4 A2

Fig. 9 it is demonstrated in [145], that, under certain conditions, the polyhedron obtained is bent in the class of polyhedrons of the same combinatorial structure but with variable faces. It is interesting to note that the volume of the polyhedron is changed under such bending . . . . . . . . The theme of flexible polyhedrons can also be found in [119] and [120]. In [119], a criterion is established, guaranteeing the possibility of stretching a polyhedron over a given polygonal line in R 3, when this polyhedron is isometric to a plane polyhedron given beforehand. It allows one to realize any polyhedral development in /P. For this purpose, it should, be cut into polygonal domains (with all of the vertexes lying on the boundaries of the cuts). Then, the cuts should be juxtaposed to the polygonal lines in R 3, and each polygonal domain should be realized. As this takes place, the transition of an edge into a link of the polygonal line is not guaranteed or required. Thus, the faces of the polyhedron obtained by such realization do not necessarily correspond to the faces of the given polyhedral metric. In principle, this work gives another variant of the theorem from [21] advocating that any development may be realized in R 3 if its sufficiently fine triangulation is admitted, and thus the face structure is not fixed. By choosing the polygonal lines (boundaries of the polyhedron parts) as variables, the flexibility of the constructed polyhedrons is shown in [120]. Moreover, this flexibilty does not guarantee the rigidity of the faces. Bending of convex polyhedrons with a boundary in the same class of convex polyhedrons has been studied in [138] and [139]. The first paper is devoted to the investigation of convex polyhedrons homeomorphic to an open circle and bounded by a simple closed polygonal line. An exhaustive description of the set of inflexible polyhedrons of the given class is presented here. The second paper presents sufficient conditions of the inflexibility or, conversely, of the flexibility of polyhedrons with a nonsplitting cut. Finally, we shall note [17], where a plane polygonal disk is continuously deformed so that the corresponding field of the second-order infinitesimal bendings has a preassigned value. This problem bears a resemblance to the well-known problem where the first variation 6g of the metric g of a surface has a preassigned value h. The case of continuous curving of a plane domain as well as the case of deformation of a plane polygon as of a polyhedron with marked vertexes are considered in [17]. The result of this work consists in the establishment of the equation for determining the first-order field of infinitesimal bendings by a given field of second order and vice versa.

8.5.

Inflexible and Infinitesimally Rigid Polyhedrons

Although almost all polyhedrons are inflexible, until recently, there has not been known any extenslve class of inflexible or infinitesimally rigid polyhedrons except the convex ones. The first examples of such polyhedrons are due to Stoker [211]. As has been mentioned above in i.8.2., Stoker has used an additional supposition - - that the spherical image of faces forms a convex quadrangle on the sphere (this is necessary for 1028

Fig. 10 applicability of Cauchy's lemmas) - - for investigating polyhedrons with nonconvex, so-called saddle vertexes in which exactly four faces come together forming two "hollows" along the edges OA2 and OA4 and two "ridges" along the edges OA1 and OA3 (Fig. 9). It has been proved that, under these conditions, any two such polyhedrons with identical combinatorial structure, similar type of corresponding vertexes, and the same planar angles on the corresponding faces will be isogonal, i.e., will have equal dihedral angles at the corresponding edges. With the addition of any further metric condition (for example, equal areas of faces, equal perimeters of faces, equal lengths of edges, etc.) to the aforesaid, the congruence of the polyhedrons is obtained. An example of a polyhedron that is nonconvex as a whole but one that is uniquely determined by this theorem is presented in Fig. 10. By removing one or several faces and gluing the resulting holes by the polyhedrons of the indicated class, the uniquely determined polyhedrons of any topological type can be obtained. Polyhedrons with boundary are also studied in [211] with the establishment of certain signs of their isogonality and congruence. Note that only angles are involved in the main lemmas and theorems considered in [211] and, consequently, the following question on the inversion of Cauchy's Theorem was posed in [211]: will the faces have equal planar angles at the corresponding vertexes if two convex isomorphic polyhedrons have equal dihedral angles? Partial positive answers may be found in [26, 59, 185]. A new approach to the problem of nonflexibility has been offered in [155], with a detailed description in [156]. This new concept implies that a space distance x between vertexes (or a number of such distances) is added to the known lengths of the polyhedron edges with choosing the pair of vertexes so that the positions of the polyhedron vertexes are afortiori exactly determinable under this additional connection. Under the alteration of x, the position of the vertexes is changed, and, under the condition of absence of the boundary, the given lengths of the edges and x satisfy a certain algebraic equation. This equation may formally be extended over the whole complex plane and then, considering the corresponding Riemann surface, it may be determined whether this equation may be fulfilled identically with respect to x or if it is in fact fulfilled only for some values of x. In the first case, we have a flexible polyhedron', and in the second case, the polyhedron is nonfiexible. Thus, it has been shown by the same method in [155] and [156] that if a suspension (a bipyramid) is embedded, it is inflexible. Unfortunately, the application of such a method implies a knowledge of the specific properties of the combinatorial structure of the polyhedron. Another class of nonflexible polyhedrons, namely, pyramids, is considered in [112]. The most interesting fact here is not the inflexibility in itself (it can be easily proved), but the existence of pyramids in any topological class. An ~xample of a pyramid of genus 1 is presented in Fig. 11. The polyhedrons generalizing the polyhedrons with single-valued projection on a certain plane are studied in [184]. Let there be a certain direction I with which all the external normals of the faces (coordinated with the orientation of the faces) make an acute angle. Such polyhedrons may have no single-valued projection on the plane and may belong to any topological class. It has been demonstrated in [184] that such polyhedrons (with specified structures) are infinitesimally rigid under the boundary condition of sliding perpendicularly to l, and that by using the projective invariance the rigidity signs may already be obtained for the polyhedrons without boundary and of any topological class. 1029

Fig. 11 8.6.

F r a m e w o r k s and Tension F r a m e w o r k s

Since the seventies, a large number of works have appeared, devoted to the investigation of kinematics and statics of frameworks and the linear realizations of graphs in /F', n > 2. This theme embracing the investigation of frameworks, polyhedrons, architectural forms, and engineering mechanics became the subject matter of a special journal "Structural Topology" founded in 1979 and published by Montreal University in English and French. Let us briefly review some works on this theme. The works [169, 163, 164, 200] can be recommended as introductory papers offering an explanation of the main terminology and formulation of the essential results, methods, and problems in this field. Papers [167] and [217] are review articles. The inflexibilityof frameworks in the general position is studied in [187], while the case of a plane is considered in [142, 143, 201]. An algorithm for checking the inflexibility of a framework on a plane is presented in [191]. The algebraic apparatus may be found in [170, 221]. Concrete types of frameworks are considered in [151,196,201,203, 223, 224]. Mechanisms as flexible frameworks are investigated in [48, 144, 240, 244]. The papers [120, 179, 216, 222] are devoted to the study of frameworks connecting bodies or containing bodies as their own elements. Various applications of the theory of frameworks are presented in [152, 168, 233]. Near the middle of the seventies, a fresh formulation of problems appeared in the theory of frameworks. This formulation is ~sociated with the highly informative and productive extension of the notion of isometric deformation, which lies in the following: a set of graph edges is subdivided into three classes, namely, bars, cables, and struts, where bars are the edges joining the vertexes which are assumed to be separated by an invariable distance, cables are the edges joining the vertexes separated by a distance which cannot be increased, and struts are the edges joining the vertexes separated by a distance which cannot be decreased. In infinitesimal deformations the corresponding velocities of the deformation zi in the vertexes pi satisfy the following relations: - pj)(z

1030

-

zj) = 0

if the edge

(Pi,Pj)

is a bar; (p, -

-

<

0

(pl - pj)(z

- zj) _> o

if the edge (p/,pj) is a cable; if the edge (pi,Pj) is a strut; The high informative value of this theory is obvious from the viewpoint o f structural mechanics. It is also quite clear that, since the cables correspond to the resistance of the edge to an extension and the struts imply the compressive strength of the edge, the introduction of cables and struts reflects the stress state of the system. In line with the preceding, a graph in R a where all of the edges are bars is called a framework, and a graph containing cables and struts is named a tension framework. It would appear reasonable that the energy of the internal stress state of the system would be taken into account in the tension frameworks, in the possibility of their deformation or, on the contrary, in their stability. The terminology here is predominantly the same as that in the theory of frameworks: rigidity of the first and second order, static rigidity, stresses and self-stresses. In addition, there is the notion of proper-stress used in cases where the stresses of the cables are nonnegative while those of the struts are nonpositive. The term "strictly proper-stress" is used in cases where the proper-stress of each cable and each strut is different from zero. In m a n y papers, the relation between the first-order rigidity and the type of self-stress is determined. We do not have t h e opportunity to mention all the reported results here and thus we shall cite only one of the most recent results. The notion of pre-stress stability is introduced in [168]. A stressed framework is called a pre-stressed stable one if it permits strictly proper-self-stresses and some specific function of energy, defined on the framework configurations, has a rigorous local minimum on the given framework. In turn, the energy function H*(q) of the framework's given configuration q is defined by the following expression:

H'(q) = ~_, fij(Iq,- qjl2),

(i,j) e E,

where flj is a strictly increasing function on the cables (i,j), and f~j is a strictly decreasing function on the struts (i,j), whereas on the bars fij has a rigorous minimum. The following principle of energy is determined with respect to the energy function introduced above: if H*(q) attains a rigorous local minimum on the configuration p, then the framework corresponding to this configuration is inflexible. Further, in [168] the definition of the second-order infinitesimal bending, which adequately reflects the properties of the bar, cable, and strut, is introduced, and the following inclusions are proved: first-order rigidity =~ pre-stress stability =~ second-order rigidity =~ inflexibility. None of the inverse inclusions is valid. Thus, among other specific results obtained in [168], there is the definition of a framework property intermediate between the first-order and second-order rigidities, where the notion of the second-order rigidity appears to be included in the system of mechanical interpretations of the behavior of the tension framework. The following works are devoted to the tension frameworks: [161,162, 167, 181,202, 217, 221,222, 233].

LITERATURE

CITED

1. A. D. Alexandrov, "On infinitesimal bendings of irregular surfaces," Mat. Sb., 1, No. 3, 307-322 (1937). 2. A. D. AIexandrov, "On one class of closed surfaces," Mat. Sb., 2,; No. 1, 69-77 (1978). 3. A. D. Alexandrov, Convex Polyhedra [in Russian], GITTL, Moscow (1950). 4. A. D. Alexandrov and E. P. Sen'kin, "On the inflexibility of convex surfaces," Vestn. LGU, No. 1,

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