411. Seiichi Yamaguchi, "On some transformations in locally product Riemannian spaces," Tensor, 18, No. 2, 227-238 (1967). 412. Seiichi Yamaguchi, "On infinitesimal projective transformations in non-Riemannian recurrent spaces," Tensor, I_~8, No. 3, 271-278 (1967). 413. Seiiehi Yamaguchi and Kozi Matumoti, "Notes on infinitesimal affine transformations in AKn-spaces," Tru Math., 2, 42-45 (1966). 414. Seiichi Yamaguchi and Kozi Matumoti, "On R i c c i - r e c u r r e n t s p a c e s , " Tensor, 19, No. 1, 64-68 (1968). 415. Hiroshi Yasuda, "Subspaces of s e m i s i m p l e group spaces. II," T e n s o r , 1__33,180-185 (1963). 416. S. T. Yau, " R e m a r k s on c o n f o r m a l t r a n s f o r m a t i o n s , " J. Different. Geom., 8, No. 3, 369-381 (1973). 417. Shinsuke Yorozu, "Notes on h a r m o n i c t r a n s f o r m a t i o n s , " Tohoku Math. J., 24, No. 3, 441-447 (1972).

BENDING E.

OF P.

CONVEX

SURFACES UDC 514.772

Sen'kin

A s u r v e y of r e s u l t s on bending of n o n r e g u l a r convex s u r f a c e s is p r e s e n t e d . INTRODUCTION In g e o m e t r y one studies three f o r m s of bendings. 1) An i s o m e t r i c map of one surface onto another, i . e . , a o n e - t o - o n e bicontinuous c o r r e s p o n d e n c e of s u r f a c e s under which the c o r r e s p o n d i n g c u r v e s on the s u r f a c e s have the same length. 2) Continuous bendings under which a surface is imbedded in a continuous family of s u r f a c e s i s o m e t r i c to it. 3) Infinitely small bendings under which a surface is d e f o r m e d in dependence on some p a r a m e t e r while the lengths of c u r v e s on the surface are stationary, i . e . , the derivative of the lengths of c u r v e s at the initial m o m e n t is equal to z e r o . A bending is called trivial if it r e d u c e s to a motion o r to a motion and a reflection. If one is concerned with infinitely small bendings, then it is called trivial in the case when at the initial moment it r e d u c e s to an ,; infinitely s m a l l motion. If a surface does not admit any infinitely small bendings except trivial ones, then it is called rigid. The problem of bending of surfaces was the source of many investigations of the geometers of the 19th and 20th centuries. The majority of these investigations are devoted to the question of bending a surface "in the small," or the investigation of special forms of surfaces. Uponpreserving the regularity of a surface, a bending is a deformation of the surface under which the line element of the surface is preserved. ~ This leads to differential equations with respect to the coordinates of points of the transformed surface. In the case of infinitely small bendings, these equations are linear. Thus, in the case of regular surfaces the question reduces to existence and uniqueness theorems for solutions of the corresponding differential equations. One of the most general results concerning bendings of surfaces "in the small" was obtained by Darboux. Darboux proved that a sufficiently small neighborhood of an arbitrary point of an analytic surface admits bending. Levi strengthened this result, proving that two sufficiently small isometric pieces of analytic surfaces with nonzero Gaussian curvature can be continuously bent so as to carry one into the other (with an additional reflection if the Gaussian curvature is greater than zero). If one removes the requirement that the Gaussian curvature be different from zero, then such a result does not hold. Efimov proved that there exist analytic surfaces containing isolated points, where all curvatures of normal sections vanish, and not admitting continuous bendings even for arbitrarily small neighborhoods of these points [11, 12, 13]. T r a n s l a t e d f r o m Itogi Nauki i Tekhniki. P r o b l e m y Geometrii, Vol. 10, pp. 193-224, 1978.

0090-4104/80/1403- 1287 $07.50 9 1980 Plenum Publishing Corporation

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The m a t t e r is different for bendings of s u r f a c e s "in the l a r g e . " The f i r s t t h e o r e m on bending of s u r f a c e s "in the l a r g e " is due to Cauchy [2]. In 1813 Cauchy p r o v e d that two closed convex potyhedra identically constituted f r o m equal f a c e s a r e equal. This c l a s s i c a l r e s u l t was f o r a long t i m e aside f r o m the differential g e o m e t r i c theory of s u r f a c e s . A t the p r e s e n t t i m e when the question is p o s e d m o r e b r o a d l y , i . e . , bendings of g e n e r a l convex s u r f a c e s a r e c o n s i d e r e d , it is n a t u r a l l y included in the g e n e r a l complex of t h e o r e m s on bendings of s u r f a c e s "in the l a r g e . " In 1941 Olovyanistmikov gave a decisive generalization of C a u c h y ' s t h e o r e m , proving that any closed convex s u r f a c e i s o m e t r i c with a closed convex polyhedron:, is itself equal to this polyhedron [17]. In 1838 Minding stated the conjecture that the s p h e r e is unbendable. But only in 1899 Liebmann and Minkowski, by different m e t h o d s , p r o v e d that the s p h e r e does not a d m i t n o n t r i v i a l i s o m e t r i c m a p s . After this the question of unbendability of closed convex s u r f a c e s occupied H i l b e r t , B l a s c h k e , L i e b m a n n , Weyl, and Cohn-Vossen. As a r e s u l t of t h e i r w o r k , t h e r e e m e r g e d a p r o o f of the fact that a closed r e g u l a r convex s u r face of positive Gaussian c u r v a t u r e does not a d m i t continuous bendings and is rigid. In 1927 Cohn-Vossen p r o v e d that i s o m e t r i c , 3 t i m e s continuously differentiable closed c o n v e x s u r f a c e s of positive Gaussian c u r v a t u r e a r e equal [14, 15]. In 1943 Herglotz lowered the r e q u i r e m e n t of r e g u l a r i t y to twice dffferentiability and e l i m i n a t e d the r e q u i r e m e n t of positive Gaussian c u r v a t u r e [11]. Cohn-Vossen is also r e s p o n s i b l e f o r the f i r s t e x a m p l e of a nonrigid closed s u r f a c e [14, 15], E s s e n t i a l l y new p o s s i b i l i t i e s in the theory of bendings opened in connection with the investigations of A l e k s a n d r o v . In 1941 A l e k s a n d r o v p r o v e d the e x i s t e n c e of a closed polyhedron r e a l i z i n g a polyhedral m e t r i c given on the s p h e r e , and this p a p e r p o s e d a fundamentally new synthetic theory of convex s u r f a c e s [2]. The new methods f o r m u l a t e d by A l e k s a n d r o v allowed Olovyanistmikov in 1941 to p r o v e the bendability of infinite convex s u r f a c e s with total c u r v a t u r e l e s s than 27r, and to find all of t h e i r bendings [18]. Aleksandrov f i r s t (in 1936) investigated infinitely s m a l l bendings of n o n r e g u l a r s u r f a c e s [3]. Writing the equations for the speed of this bending under v e r y g e n e r a l hypotheses and c a r r y i n g out the a n a l y s i s f o r a r b i t r a r y convex s u r f a c e s , A l e k s a n d r o v proved the following t h e o r e m : a convex s u r f a c e of rotation is rigid if and only if it does not contain flat p i e c e s and its s p h e r i c a l image o r its c l o s u r e c o v e r s the e n t i r e s p h e r e . In 1938 A l e k s a n d r o v c o n s i d e r e d closed s u r f a c e s which he called T - s u r f a c e s (of t o r a l type) [4]. s u r f a c e s a r e defined by the following intrinsic p r o p e r t y .

These

On a T - s u r f a c e t h e r e a r e domains of positive and negative c u r v a t u r e , s e p a r a t e d by p i e c e w i s e smooth c u r v e s , and the total c u r v a t u r e of the domains of positive c u r v a t u r e is equal to 4-~r. Despite the intrinsic geom e t r i c c h a r a c t e r of these r e q u i r e m e n t s , A l e k s a n d r o v succeeded in drawing strong conclusions about s u r f a c e s of type T: 1) On any T - s u r f a c e the domains of positive c u r v a t u r e f o r m a connected domain which is p a r t of a closed convex s u r f a c e ; t h e boundary o f this domain c o n s i s t s of p l a n a r c u r v e s . T - s u r f a c e s can be of any genus p > 0 (homeomorphic with a s p h e r e with p handles). 2) An analytic T - s u r f a c e does not admit a n o n t r i v i a l i s o m e t r i c m a p (in the c l a s s of analytic s u r f a c e s ) . 3) An analytic T - s u r f a c e is rigid. One of the v e r y s t r o n g methods of investigation of bendings of convex s u r f a c e s is the pasting method of A l e k s a n d r o v . He gave g e n e r a l conditions under which f r o m p i e c e s of convex s u r f a c e s one can p a s t e t o g e t h e r a (closed o r infinite) convex s u r f a c e [1]. This "pasting t h e o r e m " leads to m a n y t h e o r e m s on bendability of convex s u r f a c e s which one cannot get by other m e t h o d s . In all the p r e c e d i n g t h e o r e m s on bendiags of convex s u r f a c e s , an e s s e n t i a l rote is played by supplem e n t a r y r e g u l a r i t y r e q u i r e m e n t s on the s u r f a c e . In 1949 Pogorelov p r o v e d the unique determination of closed convex s u r f a c e s of bounded specific c u r v a t u r e , of convex caps in the c l a s s of c a p s , of a wide c l a s s of infinite convex s u r f a c e s with total c u r v a t u r e 21r and of infinite convex s u r f a c e s with total c u r v a t u r e l e s s than 2~r, under some additional h y p o t h e s e s a r i s i n g n a t u r a l l y f r o m the c o r r e s p o n d i n g t h e o r e m of Olovyanishnikov [20]. In a publication of 1952 Pogorelov gave a complete solution of the unique determination p r o b l e m s for g e n e r a l closed convex s u r f a c e s ; he also p r o v e d the unique determination of infinite general convex s u r f a c e s under the hypotheses f o r m u l a t e d above [21, 25].

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In 1959 Pogorelov got decisive r e s u l t s also in the t h e o r y of infinitely small bendings of genera[ convex s u r f a c e s [22, 25]. In p a r t i c u l a r , he proved that a general closed convex surface not containing fiat domains is rigid. Now if the surface contains flat domains, then it is rigid outside these domains. He proved a very general existence t h e o r e m for infinitely small bendings of convex s u r f a c e s with boundary and established the a r b i t r a r i n e s s with which these infinitely small bendings a r e constructed. He proved that a r e g u l a r surface with positive Caussian c u r v a t u r e does not admit infinitely small bendings other than r e g u l a r ones. In p a r t i c u lar, an analytic surface admits only analytic infinitely small bendings. P o g o r e l o v also c o n s i d e r s bendings of convex s u r f a c e s in spaces of constant c u r v a t u r e and Riemannian s u r f a c e s [23, 24, 25]. In 1968 Volkov got an estimate for the change in the spatial f o r m s of a general convex closed surface depending on the change in its m e t r i c [10]. F r o m this r e s u l t follows the unique determination of general closed convex s u r f a c e s . The question of bendings of r e g u l a r s u r f a c e s is well c o v e r e d in [8, 11, 19, 14, 7, 9]. tn the p r e s e n t p a p e r we are c o n c e r n e d only with bendings of n o n r e g u l a r convex s u r f a c e s . The exceptions are Secs. 2 and 4 of Chap. II, which are r e m a r k s on simplifications of the proofs for the case of r e g u l a r s u r face s. CHAPTER BASIC 1.

Some F a c t s

of t h e

T h e o r y of S u r f a c e s

I

LEMMAS "in the L a r g e "

in the present section some results of the theory of surfaces "in the large" and the theory of differential equations, which will be used in what follows, are formulated. 1. them be gruence One has

Suppose given two regular convex surfaces F I and F 2. Let an isometric correspondence between established. Points x I ~ F1 and x 2~ F 2 corresponding under the isometry are called points of conif the curvature of all normal sections in directions which correspond under the isometry are equal. the following assertion.

On isometric regular (3 times differentiable) convex surfaces of strictly positive Gaussian curvature, points of congruence lie isolatedly [14, 15]. An analogous assertion also holds for infinitely small bendings. Namely, one says that a point of a regular convex surface is stationary, if the curvatures of all normal sections at this point are stationary. On regular (thrice differentiable) convex surfaces of strictly positive Gaussian curvature, stationary points lie isolatedly. 2. Aleksandrov's Theorem. The differential equation ~ = 0 of Monge -Ampere type of elliptic type cannot have two different solutions whose difference in some domain does not change sign, but vanishes somewhere inside it [5]. 3. Pogorelov's Basic Lemma. Let F :z = z(x, y) be a general convex surface, not containing planar domains, which projects single-valuedly onto the x, y plane. Let ~(x, y) be the component along the Z axis of its bending field. Then the surface Z = ~(x, y) has negative curvature in the sense that it is impossible to pass through any point P of this surface a plane ~ such that all points of the surface close to P except P itself would be outside the plane ~, i.e., would lie on one side of the plane ~ [22, 25]~ 2.

Basic

Lemmas

Let F be a convex s u r f a c e . One says that the surface F is visible from the point O on one side if each r a y issuing f r o m the point O to any point of the surface i n t e r s e c t s F only in one point (rays going to points on the boundary of the surface m a y glide along the surface). If in addition the surface F is convexly a c c e s s i b l e f r o m the point 0 , then one says that it is visible from the point O f r o m within. In p a r t i c u l a r , a closed convex surface is visible f r o m within from each of its points lying on it. 1289

LEMMA 1 (onimbeddings) [29]. Let F l and F 2 be general i s o m e t r i c convex s u r f a c e s , visible f r o m points O1 and 02 on one side and a c c e s s i b l e convexly on one side with r e s p e c t to the points 01 and 02. Let LI and L~ be the boundaries of the s u r f a c e s F 1 and F 2 (if the s u r f a c e s a r e closed, then it will be a s s u m e d that the bounda r i e s reduce to points O1 and 02 c o r r e s p o n d i n g under the i s o m e t r y ) . Let there exist planes 1~1 and P2 p a s s i n g through the points O2 and 02 such that the s u r f a c e s F 1 and F 2 lie with r e s p e c t to Pl and P2 in one half space. Then if the distance f r o m the points 02 and O2 to points of the boundaries L 1 and L 2 which c o r r e s p o n d under the i s o m e t r y are equal, then the s u r f a c e s F 1 and F 2 are e i t h e r equal o r can be brought to such a position that the following conditions are satisfied: a) Some points x 1E F 1 and x 2 E F 2 which c o r r e s p o n d under the i s o m e t r y coincide. b) The distances from some point O of the space to the points x 1 and x 2 will be equal. e) Neighborhoods of the points x 1 and x 2 will be visible unilaterally f r o m the point O and will be s i m u l taneously convexly a c c e s s i b l e from the point O o r toward it. d) In neighborhoods of the points x 1 and x2 for any points x ~FI, x E F 2 which c o r r e s p o n d under the i s o m e t r y , one has rl(x) < r2(x), where rl(x) and r2(x) , r e s p e c t i v e l y , are the distances f r o m the point O to the points x E F 1, x E F 2 which c o r r e s p o n d under the i s o m e t r y . Roughly speaking, in neighborhoods of the points x 1 and x2 the s u r f a c e s are as though imbedded in one another. This idea was e x p r e s s e d by Cohn-Vossen [14]. Namely, Cohn-Vossen stated that the unique d e t e r m i n a tion of general ovaloids can be p r o v e d in the following two steps. 1. P r o v e that if i s o m e t r i c ovaloids a r e not equal then one can find points in them which c o r r e s p o n d under an i s o m e t r y such that for them matching small neighborhoods of these points can be imbedded in one another. 2.

Prove that such an imbedding is impossible for i s o m e t r i c ovaloids.

In our c a s e , under the hypotheses of L e m m a 1, such an imbedding is r e a l i z e d in the sense of the distance from some point. The impossibility of such an imbedding will be proved below. F o r r e g u l a r s u r f a c e s the proof of the impossibility of the imbedding can be r e a l i z e d by s e v e r a l methods (depending on the degree of r e g u l a r i t y of the surface). F o r general convex s u r f a c e s the proof is b a s e d on P o g o r e l o v ' s B a s i c L e m m a ( P a r a g r a p h 3 of Sec. 1). We prove L e m m a 1. Let F l and F 2 b e s u r f a c e s satisfying the hypotheses of L e m m a 1, and L 1 and L2be the sets of their boundary points. It will be a s s u m e d that F 1 and F2 are convexly a c c e s s i b l e f r o m OI and 02. The proof in the case when F 1 and F 2 a r e convexly a c c e s s i b l e to O[ and 02 is s i m p l e r and is c a r r i e d out analogously. We c o n s i d e r the distance r2(x) f r o m the point 02 to the point x ~ F 2. The distance f r o m the point O1 to the point c o r r e s p o n d i n g to the point x under the i s o m e t r y will be denoted by Rl(x). The following two c a s e s are possible: I) ri(x) = r2(x) for al[ x EF2; 2) on F 2 one can find a set G on which r1(x) > r2(x) or r1(x) < r2(x). If the first case holds, then as is well known, the surfaces are equal. We consider the second case. F o r definiteness it will be assumed that on G one has r1(x) > r2(x), and on the boundary of G one will have r1(x) = r2(x). By the hypothesis of the [emma, through the point 02 passes a plane P2, with respect to which F 2 lies on one side. Let L be a ray perpendicular to the plane P2 and directed to the side of the surface. The point 02 will be moved along the ray L. Then all the distances r2(x) will increase and hence the domain G will contract. At the initial moment the domain G is visible from within from the point 02. Let M 2 be a closed set containing G, on which rl(x) >- r2(x). We shall show that as soon as the point 02 moves along the ray L, at no point x 2 of the set M2will the ray O2x2touch the surface F2, and hence the set M2will be constantly visible from within from the point 02. For the proof, we imagine the point O2moving along the ray L from position O~ to position O~. Since here the distances r2(x) increase, the set M2 shrinks. The set M~' corresponding to the position of the point ! . O~, is contained inside the set M~, corresponding to the position of the point O~. Assuming that M2 is visible everywhere from within, except possibly at points for which the rays from O~ intersect its boundary, we show

1290

t h a t M 2 i s v i s i b l e f r o m 02 f r o m w i t h i n e v e r y w h e r e .

L e t x 2 b e an a r b i t r a r y p o i n t of the s e t M~, so t h a t

O,xl>O2"x2.

(1)

A t the p o i n t x 2 we c o n s i d e r the t a n g e n t c o n e . It i s n e c e s s a r y to p r o v e t h a t it d o e s n o t i n t e r s e c t the s e g m e n t O202. ' " For this it suffices to prove that if it intersects the ray L beyond the point O~ in some point B then the point B also lies beyond the point O2'. Through the points O~',02,' x 2we draw a plane. In its intersection with the surface F 2we get an arc x2A2, touching the segment x2B and going to the point A2 closest to x2, which lies on the boundary of the set M~. Since A2 lies on the boundary of the set M2, one has

02"A2=O,A,.

(2)

Since the surface F 2 is convex and by hypothesis M~ is visible from within from the point O~, except possibly for points whose rays intersect the boundary, the arc x2A2 will be convex and convexly accessible from the point O~. Together with the segment O~A2 it forms a convex curve O~A2x2. If the point O~itself lies on the boundary of the surface, then it is possible that O~ = A2 and the segment O~A2 reduces to a point. Let Aix ! be the line on the surface F i corresponding under the isom#try to the arc A2x2 so that lngth. (A,x,) = lngth. (A 2x2).

(3)

lngth. ( 02"A2x~) = lngEh . ( OiA ix,).

(4)

Since O2A ' 2 = OIAI, one has

t

We s h a l l s h o w t h a t the p o i n t 02 c a n n o t lie on the t a n g e n t x 2 B . L e t us a s s u m e the c o n t r a r y . Then the p o i n t A 2 a l s o m u s t lie on x2B a n d lngth (O~A2x 2) = 0~x 2. Since x2EM~' , one h a s O l x l -> O~x2. B u t lngth. (01A,xt) >~Olxl>~O2'x2 = lngh . (O2rA2x2) 9 I

t

By v i r t u e of (4), we c o n c l u d e t h a t O l x l = O2x2. B u t then O~x 2 <- Olx 1 = O2x2, w h i c h c a n n o t b e , s i n c e in the t r i tt T t a n g l e 02x202 the a n g l e a t the v e r t e x 02 i s o b t u s e . We d r a w t h r o u g h O~ a line p e r p e n d i c u l a r to the r a y L ; let C b e the p o i n t of i n t e r s e c t i o n o f t h i s line with the t a n g e n t x2B. O b v i o u s l y ,

O(C
(5)

while ingth (O~C) ~ 0, since by what has been proved the point" 02' does not lie on the tangent x2B. Since the line O~A2x2 is convex, one has lngth. (02~C + Cx2) >t- Ingth. (02'A2x2).

(6)

It f o l l o w s f r o m (5) and (6) t h a t

Bx2 = lngth (BC) + lngth (Cx2) > lugth. (02 tA2X2), w h e n c e , b y v i r t u e o f (4), we have Bx2>lngth. ( O,A lx, ) >~01xL>Ou" x2. The i n e q u a l i t y B x 2 > O~x 2 m e a n s p r e c i s e l y t h a t the p o i n t B l i e s on the r a y L b e y o n d the p o i n t O~. S i n c e upon d i s p l a c e m e n t of the p o i n t 02 a l o n g L the d i s t a n c e r2(x) i n c r e a s e s , at s o m e m o m e n t it t u r n s o u t t h a t the d o m a i n w h e r e r l ( x ) > r2(x) d i s a p p e a r s . T h i s p o s i t i o n of t h e p o i n t 02 we d e n o t e by O~. A t t h i s m o m e n t on s o m e s e t M one w i l l have r l ( x ) = r2(x) and e v e r y w h e r e o u t s i d e rl{x) < r2(x). Since the c l o s e d s e t M i s v i s i b l e f r o m within f r o m O~, s o m e n e i g h b o r h o o d c o n t a i n i n g it i s v i s i b l e f r o m w i t h i n f r o m O~ a l s o . The p o i n t s of the s e t M w i l l b e c a l l e d i m b e d d i n g p o i n t s a n d the p o i n t O~ w i l l be c a l l e d the p o l e . We m o v e the p o i n t 02 f r o m t h e i n i t i a l p o s i t i o n on the r a y L b y a s u f f i c i e n t l y s m a l l a m o u n t to the p o s i t i o n O~. T h r o u g h the p o i n t O[ w i l l p a s s i n f i n i t e l y m a n y p l a n e s with r e s p e c t to w h i c h the s u r f a c e F 2 i s s i t u a t e d on one s i d e . Now we can m o v e the p o i n t O2 f r o m the p o s i t i o n O~ in a whole cone K of d i r e c t i o n s . F o r e a c h d i r e c t i o n t h e r e e x i s t s a unique p o l e and s o m e s e t of i m b e d d i n g p o i n t s c o r r e s p o n d i n g to t h i s p o l e . The s e t of p o l e s c o r r e s p o n d i n g to the cone K m a k e s up s o m e s u r f a c e a. We s h a l l show that the s u r f a c e a h a s the s a m e p r o p e r t y , t h a t e a c h o f i t s p o i n t s can t o u c h a b a l i o f f i n i t e r a d i u s , i . e . , t h r o u g h e a c h p o i n t of the s u r f a c e a t h e r e p a s s e s a b a l i of f i n i t e r a d i u s s u c h t h a t a l l o t h e r p o i n t s of t h i s s u r f a c e lie o u t s i d e o r on the b o u n d a r y of t h i s ball. Ln f a c t , l e t 02 b e s o m e p o l e , a n d x 2 b e one of the i m b e d d i n g p o i n t s c o r r e s p o n d i n g to t h i s p o l e . Then O2x2 = 01xl, w h e r e x i i s the p o i n t on F 1 c o r r e s p o n d i n g u n d e r the i s o m e t r y to the p o i n t x 2. L e t O~ be an a r b i t ! t r a r y p o l e d i f f e r e n t f r o m 02. T h e n O2x 2 >_ O l x i ; t h i s m e a n s 02x2 >_ O2x 2. W h e n c e it f o l l o w s t h a t a l l p o l e s lie

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outside or on the surface of the ball of radius O2x 2 with c e n t e r at the point x 2. F r o m this p r o p e r t y of the s u r face a it follows that on it there is a smooth point (cf., e . g . , [28]). Let 02 be a smooth point of the surface a. We shall show that then the pole 02 c o r r e s p o n d s to only one imbedding point x2, i . e . , the set M consists in this case of one unique point. In fact, let us a s s u m e the c o n t r a r y : let the pole 02 c o r r e s p o n d to at least two imbedding points x 2 and x~. But then a must lie outside the ball of radius O2x2 with c e n t e r at x 2 and outside the ball of radius O2x~ with c e n t e r at x~. Consequently, the point O2 m u s t be "angular," which cannot be since 02 is a smooth point of O'. Let x I be the point on F i c o r r e s p o n d i n g under the i s o m e t r y to the point x 2. Then for the points x I and x 2 one has the equation Oix I = O2x2, and for all points x, close to x I and x 2 and c o r r e s p o n d i n g under the i s o m e t r y , one has Olx < O2x. We move the s u r f a c e s F I and F 2 simultaneously so that the segments OixI and O2x 2 should coincide. The simultaneous position of the points 02 and 02 are also the same points with which one was concerned in L e m m a 1. LEMMA 2 (on deformations of curves). Let Qi and Q2 be closed convex i s o m e t r i c curves on the hemisphere. Let the points S I and S 2 lie, r e s p e c t i v e l y , inside the domains bounded by the c u r v e s QI and Q2. Let, f u r t h e r , x I and x 2 be points on QI and Q2 which c o r r e s p o n d under the i s o m e t r y , and Six I and $2x2 be the distances f r o m them to the points Si and S 2 on the surface of the sphere. Then if for a|l points of the c u r v e s one has S i x I -< $2x2, then the c u r v e s are congruent and the points S I and S 2 in them are identically situated. LEMMA 3. Let Q2 be a planar closed convex curve and Qi be a closed curve in space which is isom e t r i c with it. Let, further, xl, Yl ~QI andx2, Y2@Q2 be points of the c u r v e s which c o r r e s p o n d under the i s o m e t r y , and r(xl, y~), r(x2, Y2) be the distances between them in space. Then if for any pair of points of the c u r v e s one has r(xl, Yl) -> r(~2,Y2),then the c u r v e s QI and Q2 are congruent. The proofs of these l e m m a s are given in [29, 30]. CHAPTER I I THEOREMS

ON BENDING OF CONVEX SURFACES WITH BOUNDARIES

1.

Bending

of General

Convex

Surfaces

In this section the following two t h e o r e m s of deformations of general convex s u r f a c e s will be p r o v e d [2830]. THEOREM 1. Let F i and F 2 be i s o m e t r i c general convex s u r f a c e s , visible unilaterally f r o m the points O1 and 02 convexly a c c e s s i b l e on one side with r e s p e c t to the points 01 and 02. Let L 1 and L2 be t h e i r bounda r i e s . Suppose through the points O1 and 02 there pass planes with r e s p e c t to which, r e s p e c t i v e l y , F 1 and F 2 are situated in one half space. Then if the distances f r o m the points 02 and 02 to points of the boundaries L 1 and L2 which c o r r e s p o n d under the i s o m e t r y are equal, then the s u r f a c e s F 1 and F 2 are equal. THEOREM 2. Let F 1 and F 2 be i s o m e t r i c general convex s u r f a c e s with a r b i t r a r y boundaries L 1 and L2, each of which does not contain r e c t i l i n e a r segments, both of whose ends lie on the boundary (segments consisting entirely of boundary points are admissible). Then if F 1 is not equal to F2, then on the boundary L 1 one can find p a i r s of points A t, B 1 and C1, D 1 such that

r(At, B,) ~>r(A2,B2), r(Cl, D,)
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F r o m t h e s e two t h e o r e m s f o l l o w s the t h e o r e m on unique d e t e r m i n a t i o n of g e n e r a l c l o s e d c o n v e x s u r f a c e s . H o w e v e r , we n o t e t h a t in t h e i r p r o o f one u s e s P o g o r e l o v T s B a s i c L e m m a (cf. S e c . 1, P a r a g r a p h 3}. In f a c t , l e t F 1 a n d F 2 b e i s o m e t r i c g e n e r a l c l o s e d c o n v e x s u r f a c e s . It i s r e q u i r e d to p r o v e t h a t t h e y a r e e q u a l . On t h e m we c h o s e two p o i n t s O1 and 02 w h i c h c o r r e s p o n d u n d e r t h e i s o m e t r y , a n d we s h a l l a s s u m e t h a t the b o u n d a r i e s c o n t r a c t to t h e p o i n t s O1 and O~. Then the s u r f a c e s F l and F 2 s a t i s f y a l l the h y p o t h e s e s o f T h e o r e m 1: a s t h e p l a n e s P l a n d P 2 o n e c a n t a k e t h e s u p p o r t p l a n e s at the p o i n t s O1 and 02, the d i s t a n c e s to t h e p o i n t s on t h e b o t m d a r y w h i c h c o r r e s p o n d u n d e r the i s o m e t r y w i l l b e e q u a l , s i n c e a l l of t h e m a r e e q u a l to zero. Consequently, the surfaces F 1 and F 2 are equal. In o r d e r to r e d u c e the t h e o r e m on unique d e t e r m i n a t i o n of g e n e r a l c l o s e d c o n v e x s u r f a c e s to T h e o r e m 2, i t s u f f i c e s to r e m o v e one at a t i m e f r o m the s u r f a c e s p o i n t s w h i c h c o r r e s p o n d u n d e r the i s o m e t r y and to a s s u m e t h a t t h e b o u n d a r i e s c o n t r a c t to t h e s e p o i n t s . Then b y T h e o r e m 2, if the s u r f a c e s a r e n o t e q u a l , one can f i n d on the b o u n d a r i e s p a i r s of p o i n t s A 1, BI a n d C1, D1, A2, B2 a n d C2, D 2 s u c h t h a t

r(Al, Bl) :>r(A~, B2), r(Cb D1) <::r(C2,D2). But t h i s c a n n o t be s i n c e a l l d i s t a n c e s b e t w e e n p o i n t s of the b o u n d a r i e s a r e e q u a l to z e r o . P r o o f o f T h e o r e m 1. L e t u s a s s u m e t h a t the s u r f a c e s F1 and F 2 a r e not e q u a l . T h e n , a c c o r d i n g to L e m m a 1, one s u r f a c e can b e " i m b e d d e d " in t h e o t h e r , i . e . , the f o l l o w i n g c o n d i t i o n s w i l l b e s a t i s f i e d : a) s o m e p o i n t s x I ~ F ~ a n d x 2 ~ F 2 w h i c h c o r r e s p o n d u n d e r the i s o m e t r y w i l l c o i n c i d e ; b) the d i s t a n c e s f r o m s o m e p o i n t o f s p a c e O to the p o i n t s x~ and x 2 a r e e q u a l ; c) n e i g h b o r h o o d s o f the p o i n t s x~ a n d x 2 a r e v i s i b l e f r o m the p o i n t O on one s i d e a n d a r e s i m u l t a n e o u s l y c o n v e x l y a c c e s s i b l e f r o m the p o i n t O o r to it; d) in n e i g h b o r h o o d s of the p o i n t s x 1 and x 2 f o r a n y p o i n t s x ~F~ a n d x ~F2 w h i c h c o r r e s p o n d u n d e r the i s o m e t r y one h a s rl{x) < r2(x), w h e r e r~(x) and r2(x) a r e the r e s p e c t i v e d i s t a n c e s f r o m the p o i n t O to the p o i n t s x ~ F 1 and x E F 2. If t h e p o i n t s x 1 a n d x 2 a r e s m o o t h , then b y v i r t u e of t h e f a c t t h a t the d i f f e r e n c e r l ( x ) - r2(x) h a s an e x t r e m u m a t x2, the a n g l e s b e t w e e n the s e g m e n t O x l a n d the d i r e c t i o n s on the s u r f a c e F 1 a t the p o i n t x l m u s t be e q u a l to the a n g l e s b e t w e e n the s e g m e n t Ox 2 a n d the d i r e c t i o n s a t the p o i n t x 2 w h i c h c o r r e s p o n d u n d e r the i s o m e t r y . If the p o i n t s x~, x 2 a r e c o n i c a l , t h e n the t a n g e n t c o n e s a t t h e m m u s t be e q u a l . In f a c t , w e l o c a t e unit s p h e r e s a t x I a n d x 2. In the i n t e r s e c t i o n s with the c o n e s we g e t two c l o s e d c o n v e x i s o m e t r i c c u r v e s Ql a n d Q2. L e t S 1 and S 2 b e the p o i n t s of i n t e r s e c t i o n with the s e g m e n t s Ox 1 and Ox 2. Then~, b y v i r t u e of the f a c t t h a t the d i f f e r e n c e r1(x) - r2(x) a c h i e v e s a m a x i m u m at x 2 , the a n g l e s f o r m e d b y the s e g m e n t Oxt with the d i r e c t i o n s on the s u r f a c e at the p o i n t x 1 w i l l b e no g r e a t e r t h a n the a n g l e s f o r m e d b y the s e g m e n t Ox 2 w i t h the d i r e c t i o n s w h i c h c o r r e s p o n d u n d e r the i s o m e t r y . But then the d i s t a n c e f r o m the p o i n t S 1 to the p o i n t s of t h e c u r v e QI on the s u r f a c e of the s p h e r e w i l l b e no g r e a t e r than the d i s t a n c e f r o m $2 to the c o r r e s p o n d i n g p o i n t s of t h e c u r v e Q~. B y v i r t u e of L e m m a 2, we c o n c l u d e t h a t Q~ and Q2 a r e e q u a l , and h e n c e the c o n e s a r e a l s o e q u a l . If a t l e a s t one of the p o i n t s x l , x 2 i s a r i d g e p o i n t , then the a n g l e f o r m e d b y the line which c o r r e s p o n d s to the r i d g e with the segznent Oxl m u s t be e q u a l to the a n g l e f o r m e d by the c o r r e s p o n d i n g line a t the p o i n t x 2 w i t h the s e g m e n t Ox2. B y m o t i o n of the s u r f a c e F~ we m a t c h the d i r e c t i o n s a t the p o i n t s x I and x 2 w h i c h c o r r e s p o n d u n d e r the i s o m e t r y (if at l e a s t one of the p o i n t s x l , x2 is a r i d g e p o i n t , then we m a t c h the d i r e c t i o n s c o r r e s p o n d i n g to t h e e d g e ) . We d e n o t e the m a t c h e d p o i n t by x o. T h e n we have r l ( x o) = r2(xo) , and f o r a l l x c l o s e to x 0 and c o r r e s p o n d i n g u n d e r the i s o m e t r y , r~(x) < r2(x). We s h a l l show t h a t s u c h an i m b e d d i n g i s i m p o s s i b l e f o r c o n v e x s u r f a c e s . with r a d i u s : , - v e c t o r Y

(x)

We c o n s i d e r the s u r f a c e F

[r~ (x) + r2 (x) ].

F o r a l l x c l o s e to x 0 it w i l l b e a c o n v e x s u r f a c e , c o n v e x l y a c c e s s i b l e on the s a m e s i d e a s the s u r f a c e s F~ and F~. The p o i n t x 0 on the s u r f a c e F c a n n o t , b y v i r t u e of the i n e q u a l i t y r~(x) < r2(x), b e l o n g to a f l a t n e i g h b o r h o o d . C o n s e q u e n t l y , it a l s o c a n n o t b e l o n g to a f l a t n e i g h b o r h o o d on F . The v e c t o r f i e l d ~- = r l - r 2 w i l l b e a b e n d i n g f o r the s u r f a c e F , s i n c e it s a t i s f i e s a L i p s c h i t z c o n d i t i o n and a l m o s t e v e r y w h e r e drdT = 0. But t h i s , a s A l e k s a n d r o v p r o v e d , is n e c e s s a r y and s u f f i c i e n t f o r T to be a b e n d i n g f i e l d f o r a g e n e r a l c o n v e x s u r f a c e [3].

1293

We introduce a Cartesian coordinate s y s t e m . We d i r e c t the z axis along the segment Ox0, and as the x, y plane we take the plane p e r p e n d i c u l a r to the segment Ox 0 and passing through the point O. We subject the surface F to the projective t r a n s f o r m a t i o n X'--

x

,

y

Z,

---F' y = ~ - '

I..~

z"

We get a convex surface F ' , which projects single-valuedly onto the x, y plane.

We subject the vector field

r(~, n, r to ~, =

~_~,

,

n

"il = "-d"

~,=

x ~ + y~ + z~ .

z

This vector field T' by the D a r b o u x - S a u e r t h e o r e m [36] wilt be a bending for the surface F ' . Since x 0 did not belong to a planar neighborhood on F, the point c o r r e s p o n d i n g to it on F ' wilt also not belong to a planar neighborhood. The radial component of the field T with r e s p e c t to the radius r will be equal to (r, + r,) (r,--r,) -'~---~

Ir,+r=l

r 21--r22 "

At the point x0, since r~ - r ] = 0, it is equal to z e r o , and at all points close to x0, less than zero. z component of the field r ' will be equal to ~,

2

2

rl--r2 Z1-1- Z=

The

9

Hence, at the point x 0 it is equal to z e r o , and at all points close to x 0 it is l a r g e r than z e r o , which is impossible by virtue of P o g o r e l o v ' s Basic L e m m a (Sec. 1, P a r a g r a p h 3). This completes the proof of T h e o r e m 1. P r o o f of T h e o r e m 2. Let us a s s u m e the c o n t r a r y . Let us a s s u m e , e . g . , that for any points x,, y~ 6L~ one has r(x l, Yi) -< r(x2, Y2). We choose on L1 an a r b i t r a r y point Or. We denote the point on L 2 c o r r e s p o n d i n g to it by 02. We c o n s i d e r the distance r2(x) from the point 02 to a point x ~ F 2. The distance f r o m the point Oi to the point c o r r e s p o n d i n g to x under the i s o m e t r y we denote by rl(x). The following three c a s e s are possible: I) ri(x) = r2(x) for all xEF2; 2) on F 2 one can find a set G, on which rt(x) > r2(x); 3) for all x ~ F 2 one has rl(x) -< r2(x). If the f i r s t case holds, then, as is well known, the s u r f a c e s are equal. We c o n s i d e r the second c a s e . Since for points of the boundaries of the s u r f a c e s Ft and F 2 one has r~(x) _< r2(x), and for points of G one has r~(x) > r2(x), on the boundary of the set G one will have rlLx) = r2(x). But then the p a r t s of the s u r f a c e s c o r r e s p o n d i n g to the sets G with the a s s o c i a t e d boundary points satisfy the hypotheses of L e m m a 1, and hence admit "imbedding." But as follows f r o m the proof of T h e o r e m 1, such an imbedding is impossible, since actually the proof of T h e o r e m 1 consists p r e c i s e l y of a proof of the i m p o s sibility of such an imbedding. We c o n s i d e r the third case. Let for all points of the s u r f a c e s F 5 and F 2 us have the inequality rl(x) r2(x) , and then we a r r i v e at case 2. Otherwise, the inequality r~(x) _< r2(x) wilt hold for all points of the surface and any positions of the points 05 and 02 on the boundaries L~ and L 2. Then we choose as the points Oi and 02 a r b i t r a r y points which c o r r e s p o n d under the i s o m e t r y of the s u r faces F t and F2, which do not lie on the boundaries. Again we consider the distances rs(x) and r(x) f r o m O5 and 02 to the points which c o r r e s p o n d under the i s o m e t r y . F o r points of the boundaries Li and L2, by virtue of what was said above, one will have r~(x) < r2(x). If one can find on the surface a point x t for which rs(x) > r2(x), then we a r r i v e at case 2 again. Otherwise, for any p a i r s of points xl, Yt 6 F l and the c o r r e s p o n d i n g pair x 2, Y26F2 one will have

r(x,, yt)
1294

(!)

We shall show that in this case the surfaces coincide locally, and then it will follow that they are also equal in the large. Let A l be a point on F1, A 2 be the c o r r e s p o n d i n g point on F 2. We shall show that the points A l and A 2 have congruent neighborhoods. We c o n s i d e r the following c a s e s : 1.

The point A l has no flat neighborhood and does not lie on a r e c t i l i n e a r segment.

2.

The point A l has a flat neighborhood.

3.

The point Ai lies on a r e c t i l i n e a r segment.

In the f i r s t case one can cut from the surface F 1 a cap @l with boundary Q1, containing the point A 1. Let Q2 be the curve on F 2 c o r r e s p o n d i n g under the i s o m e t r y to the curve Q1, and ~2 be the domain it bounds. The c u r v e s Ql and Q2, by virtue of (1), satisfy the hypotheses of L e m m a 3, and hence, m u s t be congruent. The same thing will hold for all sections not containing points of the boundary Ql of the cap r it follows already that the caps @l and @2 a r e congruent.

Whence

We c o n s i d e r the second c a s e . Let the point A 1 be planar. We choose a p l a n a r neighborhood ~l of the point A 1. Then, by virtue of (1), the neighborhood ~2 on F2, c o r r e s p o n d i n g to @l, must be planar also, and hence congruent to ~1. We c o n s i d e r the third c a s e . Let the point A 1 lie on the r e c t i l i n e a r segment l 1. Both ends of the segment I 1 by the hypotheses of the t h e o r e m cannot lie on the boundary L1. We move the support plane passing through the segment ll p a r a l l e l , negligibly into the surface F 1 to position P. In the section we get a convex figure @. Through the point A i we draw a plane, p e r p e n d i c u l a r to P and the segment l 1. Its line of intersection with the plane we denote by a . The line a divides the figure @ into two p a r t s . At least one of these p a r t s does not conrain boundary points, since by a s s u m p t i o n , at least one end of the segment II does not lie on the boundary. Hence, the plane 1D can be turned around the line a so that it cuts from the surface F l a cap @l- But then, as we have already seen, the domain @2 on F 2 c o r r e s p o n d i n g to @i will be congruent to ~t. The t h e o r e m is proved. 2.

Case

of Regular

Surfaces

F o r r e g u l a r s u r f a c e s , T h e o r e m s 1 and 2 can be proved somewhat differently and m o r e simply [5, 28]. Namely, the impossibility of imbedding, a s s e r t e d in L e m m a 1, can be proved with the help of the B a s i c Lemma of Pogorelov. .~ The proof of this lemma is complicated and occupies a large part of [22]. The rest of the proof is rather simple. Hence we n o w give two variants of the proof of impossibility of imbedding for the case of regular surfaces, depending on the degree of regularity. I. Let the surfaces F I and F 2 be twice continuously differentiable and have positive Gausgian curvature. The method which will be applied deserves s o m e attention, since here one applies the Darboux equation. This equation is very welt known and it would s e e m to be a very natural device for proving the nonbendabitity theor e m s for convex surfaces, but up to n o w no such proof has been obtained.

Let the surface F be visible from the point O unilaterally (from within or from without). W e introduce spherical coordinates r, O, q~ with center O. Let u, v be parameters on the surface so that the surface is defined by functions

r(u,v), O(u,v), ~(u,v). It is known that if the surface is twice differentiable, then the distance r(u, v) satisfies a differential equation of Monge - A m p e r e type with coefficients depending only on the line element of the surface. This Ks also the Darboux equation for r(u, v). F o r the function p = (1/2)r 2 it has the form (p,,--E) (P~2--G) -- (p,2--F)2-]-K[Epv2--2Fpup,+ GpJ - - 2 (EG--F2)p] = 0,

(I)

where PlI, Pi2, 922 are the covariant second derivatives of the function p, K is the Gaussian curvature, and E, F, G are the coefficients of the first form. If the surface has positive Gaussian curvature, then the equation will be of elliptic type.

1295

Suppose t h e r e is an i m b e d d i n g s a t i s f y i n g the h y p o t h e s e s of L e m m a 1. Then the D a r b o u x equation will have two d i s t i n c t s o l u t i o n s r~ and r2, whose d i f f e r e n c e has a m a x i m u m (minimum), equal to z e r o inside the domain of definition. But this cannot b e , by v i r t u e of A l e k s a n d r o v ' s t h e o r e m (cf. Sec. 1, P a r a g r a p h 2). 2.

L e t the s u r f a c e s F 1 and F 2 be t h r i c e d i f f e r e n t i a b l e and have p o s i t i v e G a u s s i a n c u r v a t u r e .

Suppose t h e r e is an i m b e d d i n g at the points x 1 and x2 which c o r r e s p o n d u n d e r the i s o m e t r y . in the p r o o f of L e m m a 1 it is p r o v e d that to d i f f e r e n t p o l e s c o r r e s p o n d d i f f e r e n t i m b e d d i n g p o i n t s , and infinitely many p o l e s m e a n s that on the s u r f a c e s F1 and F 2 t h e r e will be infinitely many i m b e d d i n g p o i n t s . It is e a s y to p r o v e that an i m b e d d i n g point c o r r e s p o n d s to c o n g r u e n c e p o i n t s of the s u r f a c e s F 1 and F 2. C o n s e quently, on F~ and F 2 t h e r e will be infinitely m a n y c o n g r u e n c e p o i n t s . But this c o n t r a d i c t s the fact that they m u s t lie i s o l a t e d t y (cf. Sec. 1, P a r a g r a p h 1). 3.

Infinitely

Small

Bendings

of

General

Convex

Surfaces

In this s e c t i o n two t h e o r e m s on infinitely s m a l l bendings of g e n e r a l convex s u r f a c e s , analogous to T h e o r e m s 1 and 2, wilt be p r o v e d . THEOREM 3. L e t F be a g e n e r a l s t r i c t l y convex s u r f a c e with a r b i t r a r y b o u n d a r y L. Under a n o n t r i v i a l infinitely s m a l l bending of the s u r f a c e F , one can find on the b o u n d a r y L p a i r s of points A, B and C, D such that fort= 0

r(A, B) >0, r'(C,D) <0, w h e r e r is the d i s t a n c e in s p a c e , and the dot d e n o t e s d i f f e r e n t i a t i o n with r e s p e c t to t i m e . we have the following t h e o r e m . THEOREM 4. A c l o s e d s t r i c t l y convex s u r f a c e is r i g i d . c l o s e d s u r f a c e and apply T h e o r e m 3.

From Theorem 3

In f a c t , it s u f f i c e s to e x t r a c t one point f r o m a

To p r o v e T h e o r e m 3, we r e q u i r e the following l e m m a . LEMMA 4. L e t Q be a p l a n a r c l o s e d convex c u r v e not containing any r e c t i l i n e a r s e g m e n t s . L e t x and y be a r b i t r a r y points on the curve Q. Then if the lengths of a l l a r c s of the c u r v e Q a r e s t a t i o n a r y , then f o r t = 0, f r o m ~(x, y) >- 0 it follows t h a t / : ( x , y) = 0. F o r finite bendings this [ e m m a is p r o v e d in [29]. The p r o o f for infinitely s m a l l bendings is analogous; it suffices s i m p l y to r e p l a c e finite i n c r e m e n t s by s p e e d s . P r o o f of T h e o r e m 3. L e t T be a bending field for the s u r f a c e F . L e t us a s s u m e the c o n t r a r y . Suppose f o r all x, y EL f o r t = 0 one has e i t h e r }(x, y) _> 0 o r ~(x, y) <_ 0. F o r d e f i n i t e n e s s , let us a s s u m e that on L , ~(x, y) _> 0. We choose on L an a r b i t r a r y point O. L e t x be an a r b i t r a r y point of F , and r(x) be the d i s t a n c e f r o m it to the point O. The following t h r e e c a s e s a r e p o s s i b l e (for t = 0). 1.

F o r a l l x E F one has /:(x) = 0.

2.

On F one can find a s e t G on which ~(x) < 0.

3.

F o r a l l x EF one has the i n e q u a l i t y ~(x) >- 0.

If the f i r s t c a s e h o l d s , then the field T is t r i v i a l .

We c o n s i d e r the second c a s e . Since f o r points of L one has ~(x) -> 0, and f o r points of G one has ~(x) < 0, on the b o u n d a r y of G one will have /:(x) = 0. L e t P be the s u p p o r t plane of the s u r f a c e F at the point O. A s s o c i a t i n g with the field v a suitable t r a n s p o r t in the d i r e c t i o n p e r p e n d i c u l a r to the plane P, it is e a s y to see that the r a d i a l component ~1 will be e v e r y where -> 0 and at l e a s t one i n t e r i o r point ~1 = 0. In f a c t , let a be the t r a n s p o r t v e c t o r p e r p e n d i c u l a r to P and d i r e c t e d into the half space in which F is s i t u a t e d . Its r a d i a l components at a l l points a r e p o s i t i v e . Hence one can s e l e c t a so that the r a d i a l component r l of the v e c t o r 1-1 = 9 + a will be e v e r y w h e r e >- 0 and s o m e w h e r e in the i n t e r i o r ~1 = 0. We i n t r o d u c e a C a r t e s i a n c o o r d i n a t e s y s t e m , taking as the x, y plane P. We s u b j e c t the s u r f a c e F to the p r o j e c t i v e t r a n s f o r m a t i o n x'= x,

1296

y ,= ~ ,g

z ' = ) -1.

We s u b j e c t the f i e l d 71 to the t r a n s f o r m a t i o n = z '

~'=

-~'

z

The f i e l d 72 o b t a i n e d w i l l b e a b e n d i n g f o r the t r a n s f o r m e d

s u r f a c e F 1.

The r a d i a l c o m p o n e n t of t h e f i e l d 72

i s equal to

~r The z c o m p o n e n t of the f i e l d 72 i s e q u a l to ~r z

H e n c e , f o r a l l p o i n t s of F~ one w i l l h a v e ~' > 0 a n d f o r a t l e a s t one i n t e r i o r p o i n t ~' = 0. B u t then the s u r f a c e z = ~'(x, y) w i l l have p o i n t s of s t r i c t c o n v e x i t y , w h i c h i s i m p o s s i b l e , b y v i r t u e of t h e B a s i c L e m m a of P o g o r e lov (Sec. 1, P a r a g r a p h 3). We c o n s i d e r t h r e e c a s e s . S u p p o s e f o r a l l p o i n t s of the s u r f a c e F one h a s ~(x) _> 0. We m o v e the p o i n t 0 a l o n g t h e b o u n d a r y L s o t h a t it s h o u l d r u n t h r o u g h a l l p o i n t s of L . It c a n h a p p e n t h a t one can find a p o s i t i o n O o f the p o i n t O, f o r w h i c h f o r a t l e a s t one p o i n t x one h a s ~(x) < 0, a n d then we a r r i v e at c a s e 2. O t h e r w i s e , ~(x) _> 0 w i l l h o l d f o r any p o i n t x E F a n d a n y p o s i t i o n of O on the b o u n d a r y . Now l e t t h e p o i n t O r u n t h r o u g h a l l p o i n t s of F . F o r p o i n t s of the b o u n d a r y , ~(x) -> 0. H e n c e , one can f i n d a p o i n t x E F f o r w h i c h ~(x) < 0, a n d a g a i n we a r r i v e at c a s e 2. O t h e r w i s e , f o r a l l p a i r s of p o i n t s x , y of t h e s u r f a c e F one w i l l h a v e ~(x) _> 0. We s h a l l show t h a t t h e f i e l d 7 in t h i s c a s e is l o c a l l y t r i v i a l . on the e n t i r e s u r f a c e a l s o .

W h e n c e it w i l l f o l l o w now t h a t i t i s t r i v i a l

L e t x b e an a r b i t r a r y p o i n t of F . We c u t o u t w i t h s o m e p l a n e a a c a p P c o n t a i n i n g x. The b o u n d a r y of t h e c a p Q i s a p l a n e c u r v e s a t i s f y i n g the c o n d i t i o n s of L e m m a 4. C o n s e q u e n t l y , f o r any x , y EQ one w i l l have ~(x, y) = 0. T h e s a m e h o l d s f o r a n y s e c t i o n of the c a p P n o t c o n t a i n i n g p o i n t s of Q. But f r o m t h i s it f o l l o w s now t h a t the f i e l d T is t r i v i a l on the c a p P . T H E O R E M 4. L e t F b e a g e n e r a l c o n v e x s u r f a c e with b o u n d a r y L , n o t c o n t a i n i n g p l a n a r d o m a i n s and v i s i b l e u n i l a t e r a l l y f r o m t h e p o i n t O. L e t t h e r e e x i s t a s u i t a b l e p l a n e t h r o u g h O , with r e s p e c t to w h i c h the s u r f a c e F i s s i t u a t e d in one h a l f s p a c e . Then if a l l d i s t a n c e s f r o m the p o i n t 0 to p o i n t s o f the b o u n d a r y L a r e s t a t i o n a r y , then the s u r f a c e F is r i g i d . F o r t h e c a s e of r e g u l a r s u r f a c e s t h i s t h e o r e m i s p r o v e d in [5], f o r the c a s e of g e n e r a l s u r f a c e s b y P o g o r e l o v in [22] a n d it i s a s i m p l e c o r o l l a r y of the B a s i c L e m m a . x EF.

In f a c t , l e t 7 b e a b e n d i n g f i e l d f o r the s u r f a c e F , and r(x) b e the d i s t a n c e f r o m the p o i n t 0 to the p o i n t If all ~(x) = 0, then the surface is rigid.

Otherwise one could find a set G on which either ~(x) < 0 or ~(x) > 0. For definiteness we shall assume ~(x) < 0. On the boundary of the set G one will have ]~(x) = 0. The rest of the argument coincides completely with the argument of point 2 of the proof of Theorem 3. As a matter of fact, associating with the field 7 suitable transport in the direction perpendicular to the plane P, we see that the radial component of ~(x) will be everywhere greater than zero while somewhere inside ~(x) = 0. Constructing as in point 2 a projective transformation we arrive at a contradiction with the Basic Lemma of Pogorelov (Sec. 1, Paragraph 3). 4.

Case

of

Regular

Surfaces

A s in the c a s e of f i n i t e b e n d i n g s f o r r e g u l a r s u r f a c e s , the p r o o f s of T h e o r e m s 3 and 4 can b e s i m p l i f i e d a n d the final a r g u m e n t s can b e m a d e w i t h o u t the B a s i c L e m m a and b y d i f f e r e n t m e t h o d s d e p e n d i n g on the d e g r e e of r e g u l a r i t y of the s u r f a c e . E v e r y t h i n g r e d u c e s to s h o w i n g t h a t the r a d i a l c o m p o n e n t of t h e b e n d i n g f i e l d c a n n o t have a m a x i m u m ( m i n i m u m ) e q u a l to z e r o . 1. L e t the s u r f a c e F b e t w i c e d i f f e r e n t i a b l e and have p o s i t i v e G a u s s i a n c u r v a t u r e . The v e l o c i t y ~ of c h a n g e of the d i s t a n c e r f o r an i n f i n i t e l y s m a l l b e n d i n g of the s u r f a c e s a t i s f i e s a l i n e a r d i f f e r e n t i a l e q u a t i o n w h i c h c a n b e o b t a i n e d if one t a k e s in D a r b o u x ' s e q u a t i o n r a s a function of t i m e and d i f f e r e n t i a t e s the e q u a t i o n w i t h r e s p e c t to t.

1297

F o r a surface of positive c u r v a t u r e , visible from the origin unilaterally, this equation will be of elliptic type. But as Aleksandrov showed, a linear homogeneous equation of elliptic type cannot have a nonzero solution which in its domain of definition does not change sign and vanishes somewhere inside the domain. 2. Let the surface be thrice differentiable and have positive Gaussian c u r v a t u r e . In the proof of Theor e m 4, associating with the bending field suitable parallel t r a n s p o r t , we got a point for which ~ = 0, and outside ~ > 0. Infinitely many such parallel t r a n s p o r t s can be a s s o c i a t e d . To each such t r a n s p o r t will c o r r e spond different points for which ~ = 0 and outside ~ > 0. It is easy to show that these points are stationary points of the s u r f a c e . Consequently, on the surface we get infinitely many stationary points, which c o n t r a dicts P a r a g r a p h 1, See. 1. 5.

Bending

a Convex

Surface

with

Given

Boundary

Strip

In [7] a t h e o r e m is proved about bending a r e g u l a r convex surface with boundary L with previously given boundary strip along the boundary. By a boundary strip along the boundary is meant a superficial strip along the boundary L, whose n o r m a l s are n o r m a l to the surface. It is shown that two i s o m e t r i c s u r f a c e s F 1 and F 2 with boundaries LI and L2 are equal if the superficial strips along the boundaries are flat, tangent to some sphere (where the boundaries L1 and L 2 lie on the sphere), cylindrical o r conical. Rembs p r o v e s the rigidity of a r e g u l a r convex surface under the conditions that at the initial moment of the bending the conical strip r e m a i n s conical [35]. These r e s u l t s hold also for the case of general convex s u r f a c e s . We give p r e c i s e formulations of the r e s u l t s . We consider first the c a s e s when the strip is conical and cylindrical. The c a s e s of flat and spherical strips will be c o n s i d e r e d at the end. Let F be a convex surface with boundary L. By a support plane to F at the point x 0 on the boundary L we mean a plane P passing through the point x0, with r e s p e c t to which F lies in one half space, and the plane P contains at [east one g e n e r a t o r of the tangent cone to F at the point x 0. THEOREM 5. Let F~ and F 2 be i s o m e t r i c convex s u r f a c e s with boundaries LI and L2 of bounded v a r i a tion of geodesic c u r v a t u r e . Let the support planes at points of the boundaries LI a n d L 2 P a s s through some points O~ and 02, r e s p e c t i v e l y , while F t and F 2 a r e convexly a c c e s s i b l e unilaterally with r e s p e c t to the points Ot and 02. Then the s u r f a c e s F 1 and F 2 a r e equal. THEOREM 6. Let F 1 and F 2 be convex s u r f a c e s with boundaries L1 and L 2 of bounded variation of geodesic c u r v a t u r e . Let the support planes at points of the boundaries L1 and L 2 envelop cylinders. Then the s u r f a c e s F t and F 2 are equal. Analogous rigidity t h e o r e m s hold. THEOREM 7. Let F be a convex surface with boundary L of bounded variation of geodesic c u r v a t u r e . Let the support pla~es at points of the b o u n d a r y L p a s s through some point O. We choose on L an a r b i t r a r y point x. We denote by a the angle between an a r b i t r a r y n o r m a l to the support plane at the point x and the s e g ment Ox. At the initial moment it is equal to 7r/2. Then if for an infinitely small bending ~ = 0, the surface F is rigid outside the planar domains. The dot denotes differentiation with r e s p e c t to time at t = 0. THEOREM 8. Let F be a convex surface with boundary L of bounded variation of geodesic c u r v a t u r e . Let the support planes at points of L envelop a cylinder. We denote by a the angle between an a r b i t r a r y n o r mal to the support plane at a point of the boundary L and the g e n e r a t o r of the cylinder passing through this point. At the initial m o m e n t a = ~ / 2 . Then if 5 = 0, then the surface F is rigid outside the planar domains. P r o o f of T h e o r e m 5. F o r definiteness we shall a s s u m e that the s u r f a c e s F 1 and F 2 a r e convexly a c c e s sible f r o m the points O2 and 02. We denote by K~ and K 2 cones with v e r t i c e s at the points O2 and O2 and d i r e c t o r s L1 and L 2. The a r e a s of the spherical images of the c u r v e s L1 and L 2 a r e equal to z e r o , since the spherical images of the c u r v e s L 1 and L 2 a r e the boundaries of the spherical images of the convex cones K 1 a n d K 2. The geodesic c u r v a t u r e s of any corresponding a r c s of the curves L1 and L 2 o n the sides of the s u r faces F~ and F 2 are equal by the i s o m e t r y . But since the sum of the right and left geodesic c u r v a t u r e s of a curve on a surface are equal to the a r e a of its spherical image, the geodesic c u r v a t u r e s of c o r r e s p o n d i n g a r c s of the c u r v e s L l and L 2 on the cones K l and K 2 a r e also equal. We develop the cones Kt and K 2 onto a plane, as a p r e l i m i n a r y developing them according to the g e n e r a t o r s . The c u r v e s L1 and L 2 are c a r r i e d here into plane c u r v e s Ll and L~, for which a r c s c o r r e s p o n d i n g under the i s o m e t r y will have identical geodesic c u r v a t u r e s . 1298

Since a plane curve is uniquely determined by its geodesic c u r v a t u r e , defined as a function of a r c length, the c u r v e s L~ and L~ will be congruent. Let At, B 1 and A2, B2 be the ends of the c u r v e s L~ and L~. Then the distances between the points A, B and A2, B 2 will be equal. We consider the triangles O1A1Bt and O2A2B2. They are i s o s c e l e s , since O1A1 = OrB 1 and O2A2 = O2B2. In addition, their b a s e s AtB 1 and A2B2 are equal. The angles at the v e r t i c e s 01 and 02 are also equal, by virtue of the i s o m e t r y of the surface F 1 and F 2. Consequently, the t r i a n g l e s OtA1Bt and O2A2B2 are equal. But then all the distances rl(x) and r2(x) f r o m the points O1 and 02 to points x c o r r e s p o n d i n g under the i s o m e t r y on the c u r v e s L 1' andL 2' are equal, and hence also the c u r v e s LI and L2. Now the equality of the s u r f a c e s F 1 and F 2 follows f r o m T h e o r e m 1. P r o o f of T h e o r e m 6. We denote by C~ and C 2 the cylinders which are enveloped by the support planes at the points of the boundaries Lt and L 2. Let 11 a n d / 2 be g e n e r a t o r s of these cylinders which c o r r e s p o n d under the i s o m e t r y , p a s s i n g through points x~ and x 2 on the boundaries LI and L 2 which c o r r e s p o n d under the isoraetry. We choose on /I and 12 points a I and a 2 at identical distances from the points x I and x 2 such that the surfaces F I and F 2 are visible unilaterally from them, e.g., from within. We draw through al and a 2 planes P[ and P2, perpendicular to the generators. The parts of the cylinders between the planes P[ and P2 and the boundaries L I and L 2 we develop onto a plane, developing them as a preliminary with respect to the segments xla I and x2a 2. The boundaries Lt and L 2 go into some planar curves L~ and L~, whose distances to the ends from some line c~ will be equal. The areas of the spherical images of the boundaries L! and L 2 are equal to zero, the geodesic curvatures of corresponding arcs on the sides of the surfaces are equal, hence the geodesic curvatures of c o r r e sponding arcs of the plane curves L~ and I~2 are also equal. But then the curves L~ and L~ are congruent. But since the distances to the ends of L] and L~ from some line ~ are equal, the distances from the line c~ to all points on L~ and L~ which correspond under the isometry are also equal. Consequently, for the surfaces F I and F 2 the distances to points of the boundary which correspond under the isometry from some plane are equal. But then, as is well known, the surfaces F I and F 2 are congruent [21]. The proofs of the corresponding rigidity Theorems 3 and 4 are analogous; it suffices merely to replace finite increments by velocities. Proof of Theorem 7. Let F be a convex surface and L be its boundary. We denote by K the cone which is enveloped by the support planes to the boundary L. Let co be the area of the spherical image of an are of the boundary L. At the initial moment of time, co = 0. From the condition & = 0 it follows that ~ = 0. Consequently, d) = PI + P2, where PI is the geodesic curvature of the arc of the curve L on the side of the surface, and P2 is the geodesic curvature of the same arc on the cone K. But PI = 0. Hence, f~2 = 0. Thus, the geodesic curvatures of all arcs of the curve L on the cone K are stationary. We cut the cone K along an arbitrary generator OA and we unroll it on a plane. The curve L goes into a plane curve L', for which, under infinitely small bendings the geodesic curvatures of all arcs are stationary. Consequently, L' at the initial moment of time is moved as a rigid body and the distance between its ends is stationary. We consider the triangle OAA. The velocities of change of length along its sides OA and OA are the same and equal to the velocity of change of length of the generator OA of the cone K. Now the length of the side AA is stationary. Moreover, the angle AOA is stationary, since the angle at the vertex of the cone K is stationary under infinitely small bendings. Hence, the velocity of change of the side OA is equal to zero. Then the velocity of change of all distances r(x) from the point O to points of the curve L ' , and hence also L, is equal to zero. But under these conditions, as is well kno~m, the surface F is rigid outside the planar domains.

P r o o f of T h e o r e m 8. Let F be a convex surface with boundary L. We denote by C the cylinder which is enveloped b y t h e support planes at points of the boundary L. We choose on any g e n e r a t o r of the cylinder C a point a and through it we draw a plane P, perpendicular to the g e n e r a t o r . F o r definiteness it will be a s s u m e d that the surface F is convexly a c c e s s i b l e f r o m the point a. Let w be the a r e a of the spherical image of an a r c of the boundary L. At the initial moment of time co = 0. F r o m the condition & = 0 it follows that & = 0. Consequently, P = 0, where P is the geodesic curvature of the a r c of the boundary L on the side of the cylinder C. We cut the cylinder C along the g e n e r a t o r ax, where x is a point on the boundary L through which the g e n e r a t o r p a s s e s . We unroll the part of the cylinder C between the plane P and the boundary L onto a plane. The curve L goes into a plane curve L ', for which each a r c has stationary geodesic c u r v a t u r e . Consequently, at the initial moment of time the curve L ' moves like a rigid body.

1299

We associate with a bending field of the surface F the field of parallel t r a n s p o r t along g e n e r a t o r s of the cylinder C such that the length of the segment ax is stationary. Consequently, the heights of all poirits of the curve from some line a will be stationary. But then the surface F is rigid outside the p l a n a r domains [25]. We c o n s i d e r the case when the boundary strip is planar and spherical. Let F~ and F 2 be i s o m e t r i c s u r faces with boundaries L~ and L 2. Let the support planes at points of the boundaries L1 and L 2 lie, r e s p e c tively, in one plane. Then the spherical images of the s u r f a c e s F1 and F 2 are equal to 4~. But then, as is well known, the s u r f a c e s F 1 and F 2 a r e equal. One has the c o r r e s p o n d i n g rigidity t h e o r e m . THEOREM 9. Let F be a convex surface with boundary L , such that the support planes at points of the boundary L lie in one plane P. Let ~ be the angle between the n o r m a l to any support plane at a point on L and the plane P. At the initial moment it is equal to z e r o . Then if & = 0, then the surface F is rigid outside planar domains. In fact, the boundary L is a plane curve. By virtue of the condition & = 0, it has the p r o p e r t y that the geodesic curvature of any of its a r c s is stationary. Consequently, at the initial m o m e n t of t i m e , the distance between any two points of the curve L is stationary. If one adjoins to F the p l a n a r domain bounded by the curve L , then we get a closed convex surface with stationary m e t r i c . As proved in [22], it is rigid outside planar domains. Let the boundary strip be spherical. F i r s t l y , we note that if the s u r f a c e s F~ and F 2 have boundaries L t and L 2 , lying in an open h e m i s p h e r e , then the condition of tangency of the support planes on the boundary to a sphere is superfluous. In fact, let the boundaries L 1 and L2 lie in an open h e m i s p h e r e . We c o n s i d e r the distances from the c e n t e r of the sphere to points of the boundaries L1 and L 2. which c o r r e s p o n d under the i s o m e t r y . They are all equal to one, and hence are equal. But then the s u r f a c e s F 1 and F 2 are equal. If the boundaries L1 and L 2 leave the boundary of the h e m i s p h e r e , then by virtue of the fact that the support planes on the boundary are tangent to the sphere, one gets the equality of the geodesic c u r v a t u r e s of any a r c s of the c u r v e s L~ and L 2 o n the sphere which c o r r e p o n d under the i s o m e t r y . By virtue of the fact that c u r v e s on the sphere, just as in the plane, are uniquely determined by the geodesic c u r v a t u r e s , given as functions of a r c length, it follows that L 1 and L 2 are congruent. Consequently, the s u r f a c e s F~ and F 2 are equal. The c o r r e s p o n d I n g rigidity t h e o r e m s are proved analogously. CHAPTER NONBENDABILITY 1.

Nonbendability

of Closed

Convex

OF

III

CONVEX

SURFACES

Surfaces

The p r o b l e m s of unique determination and rigidity of closed convex s u r f a c e s in t h r e e - d i m e n s i o n a l space were solved by Pogorelov [21, 22]. In the p r e s e n t p a p e r analogous r e s u l t s are established for h y p e r s u r f a c e s of n-dimensional euclidean space. F i r s t the questions of unique determination and rigidity are solved for closed convex h y p e r s u r f a c e s and then this r e s u l t is used to prove local unique determination and rigidity. THEOREM 10.

I s o m e t r i c smooth closed convex h y p e r s u r f a c e s are equal.

THEOREM 11. A closed convex h y p e r s u r f a c e not containing planar domains of dimension n - 1 is rigid. If it contains p l a n a r domains, then it is rigid outside the p l a n a r domains. THEOREM 12. Let Ft be a smooth convex h y p e r s u r f a c e and P~ be a point at which it is s t r i c t l y convex. Let F 2 be a smooth convex h y p e r s u r f a c e i s o m e t r i c with FI, and 1:)2 be the point c o r r e s p o n d i n g to P1 under the i s o m e t r y . Then sufficiently small neighborhoods of the points P~ and P2 are congruent. THEOREM 13. A convex h y p e r s u r f a c e that contains no p l a n a r domains of dimension n - 1 is rigid in the neighborhood of each point of s t r i c t convexity. If it contains planar domains, then it is rigid outside the planar domains. faces.

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p r o o f of T h e o r e m 10. F i r s t l y , we note that L e m m a 1 (on imbeddings) (Chap. I) also holds for h y p e r s u r The proof is completely analogous; it suffices simply to replace the word surface by h y p e r s u r f a c e .

L e t F 1 a n d F 2 be c o n v e x c l o s e d i s o m e t r i c h y p e r s u r f a c e s . If they a r e n o t e q u a l , then a c c o r d i n g to the [ e m m a on i m b e d d i n g s , they c a n be put in a p o s i t i o n in which the f o l l o w i n g c o n d i t i o n will hold: s o m e p o i n t s on F 1 a n d F 2 w h i c h c o r r e s p o n d u n d e r the i s o m e t r y c o i n c i d e at s o m e p o i n t Xo; the d i s t a n c e s f r o m s o m e p o i n t of the s p a c e O to at1 p o i n t s x 1 E F l, x 2 EF 2 in a n e i g h b o r h o o d of the p o i n t x 0 which c o r r e s p o n d u n d e r the i s o m e t r y s a t i s f y the i n e q u a l i t y Ox~
r=-~(r1+r2), w h e r e r 1 and r2 a r e r a d i u s v e c t o r s g o i n g to p o i n t s of the h y p e r s u r f a c e s F 1 and F~ which c o r r e s p o n d u n d e r the tsometry. The s u r f a c e ~ in a n e i g h b o r h o o d of the p o i n t x 0 will be c o n v e x , which was p r o v e d f o r the c a s e of t w o d i m e n s i o n a l s u r f a c e s in P o g o r e l o v [21]. L e t us a s s u m e that at s o m e p o i n t x* on F I , c l o s e to x0, t h e r e o c c u r s the t o s s of l o c a l c o n v e x i t y of the s u r f a c e %. L e t P j a n d P2 be t a n g e n t h y p e r p l a n e s at the p o i n t s x* on F~ a n d F 2. We denote b y n the i n n e r n o r m a l to

the hyperplane

(1/2)(P I +

P2)-

We join a point y*, close to x*, with a minimizing path T on F I. Let ri(s) be the radius vector of a point on the minimizing path T, and re(s) be the radius -vector of the corresponding point on F 2. Applying Liberman's theorem on the convexity of geodesics to the minimizing path 3/and the corresponding minimizing path on F2, we get a (r, - ~ - r 2 ) n > 0 -~f o r a l l p o i n t s of the m i n i m i z i n g path 3/. Whence it follows t h a t f o r all p o i n t s y * , c l o s e to x*, one has

[r~ (y*) --r~ (x*) + r2 ( : ) --r2 (x*) ] n>0. But t h i s m e a n s that at the p o i n t x* one has l o c a l c o n v e x i t y , which c o n t r a d i c t s the h y p o t h e s i s . We c o n s i d e r on the h y p e r s u r f a c e ~ the v e c t o r f i e l d T~rl--r2.

We choose at the p o i n t x 0 on ~ two i n d e p e n d e n t d i r e c t i o n s and we c o n s t r u c t a t h r e e - d i m e n s i o n a l p l a n e P s u c h t h a t it c o n t a i n s t h e s e d i r e c t i o n s and the s e g m e n t Ox 0. In the s e c t i o n of the h y p e r s u r f a c e ~ we get a t w o - d i m e n s i o n a l convex s u r f a c e ~. We d e c o m p o s e the v e c t o r f i e l d ~- into c o m p o n e n t s , one of which r ' t i e s in the t h r e e - d i m e n s i o n a l p l a n e P , and the r e s t a r e p e r p e n d i c u l a r to it. O b v i o u s l y the f i e l d "r' w i l l be a b e n d i n g f i e l d of a in the t h r e e - d i m e n s i o n a l p l a n e P. We i n t r o d u c e a C a r t e s i a n c o o r d i n a t e s y s t e m in the p l a n e P , t a k I n g as z a x i s the s e g m e n t Ox 0 a n d as x, y p l a n e the p l a n e p e r p e n d i c u l a r to Ox 0 at the p o i n t O. We s u b j e c t cr to the p r o j e c t i v e t r a n s f o r m a t i o n ,

-~:

x=~-,

y'=~,

1

ZP~-

z

.

We s u b j e c t the f i e l d r ' to the t r a n s f o r m a t i o n

~'~,

~'~-

r

x~+~n+z~

Z'

Z

The field ~-] o b t a i n e d will be a b e n d i n g t r a n s f o r m a t i o n of the s u r f a c e ~'. The r a d i a l c o m p o n e n t of the field T~ iS e q u a l to (r,--r~) (r, + r~)

I rlq-r2 I

2

2

rl--r2 I r, q-r2] "

1301

The z c o m p o n e n t o f the f i e l d T] i s r~

r2

2t

S i n c e at the p o i n t x 0 one h a s r l ( x 0) = r2(x0) , and in a n e i g h b o r h o o d of the p o i n t x 0 one h a s rl(x) < r2(x), we g e t ~' = 0 at the p o i n t x 0 and ~' < 0 in a n e i g h b o r h o o d of the p o i n t x 0. C o n s e q u e n t l y , the z c o m p o n e n t of the f i e l d T~ on the s u r f a c e a ' h a s a p o i n t of s t r i c t c o n v e x i t y , w h i c h c o n t r a d i c t s P o g o r e l o v ' s B a s i c L e m m a . T h e o r e m 10 i s p r o v e d . P r o o f of T h e o r e m 11. The p r o o f w i l l p r o c e e d b y i n d u c t i o n . A t w o - d i m e n s i o n a l c l o s e d c o n v e d s u r f a c e in t h r e e - d i m e n s i o n a l s p a c e is r i g i d o u t s i d e p l a n a r d o m a i n s a c c o r d i n g to P o g o r e l o v ' s t h e o r e m . L e t u s a s s u m e t h a t an (n - 2 ) - d i m e n s i o n a l c l o s e d c o n v e x s u r f a c e in (n - D - d i m e n s i o n a l s p a c e i s r i g i d o u t s i d e (n - 2 ) - d i m e n s i o n a l pLanar d o m a i n s , a n d we s h a l l show t h a t an ( n - D - d i m e n s i o n a l s u r f a c e in n - d i m e n s i o n a l s p a c e i s r i g i d o u t s i d e (n - 1 ) - d i m e n s i o n a l d o m a i n s . L e t F be a c l o s e d c o n v e x h y p e r s u r f a c e w i t h p l a n a r d o m a i n s . We c h o o s e two p o i n t s x and y n o t b e l o n g ing to p l a n a r d o m a i n s . We d r a w t h r o u g h t h e m a h y p e r p l a n e P s u c h t h a t it i n t e r s e c t s (11 - 2 ) - d i m e n s i o n a l e d g e s a t the p o i n t s x and y if t h e y e x i s t . In the s e c t i o n we g e t an (n - 2 ) - d i m e n s i o n a l cLosed c o n v e x s u r f a c e a , w h i l e t h e p o i n t s x and y w i l l n o t l i e in (n - 2 ) - d i m e n s i o n a l p l a n a r d o m a I n s . We d e c o m p o s e the f i e l d T into a c o m p o n e n t ~ ' , l y i n g in P , a n d To, p e r p e n d i c u l a r to it, w h i c h d o e s n o t e f f e c t a c h a n g e of d i s t a n c e r ( x , y ) . By v i r t u e o f the h y p o t h e s i s , a i s r i g i d in t h e p l a n e P o u t s i d e (n - 2 ) - d i m e n s i o n a l p l a n a r d o m a i n s . H e n c e , the d i s t a n c e s b e t w e e n any p a i r s of p o i n t s x a n d y on F n o t beLonging to p l a n a r d o m a i n s a r e s t a t i o n a r y . B u t t h i s m e a n s t h a t F is rigid outside planar domains. P r o o f of T h e o r e m s 12 and 13. We s h a l l p r o v e T h e o r e m 13. L e t F be a c o n v e x h y p e r s u r f a c e a n d P b e a p o i n t at w h i c h it i s s t r i c t l y c o n v e x . It i s n e c e s s a r y to show t h a t F i s r i g i d o u t s i d e p l a n a r d o m a i n s in a n e i g h b o r h o o d of the p o i n t P . W i t h a p l a n e Q , p a r a l l e l to t h e s u p p o r t p l a n e a t the p o i n t P , we cut f r o m F a s u f f i c i e n t l y s m a l l c a p w. In the s e c t i o n we g e t an (n - 2 ) - d i m e n s i o n a l s u r f a c e F ' w h i c h w i l l be r i g i d in Q o u t s i d e p l a n a r d o m a i n s . C o n s e q u e n t l y , the d i s t a n c e s r ( x , y) b e t w e e n any p a i r s o f p o i n t s on F ' , n o t b e l o n g i n g to p l a n a r d o m a i n s , w i l l be s t a t i o n a r y . L e t a t l e a s t one of the p o i n t s x , y lie in a p l a n a r d o m a i n on F ' . If it a l s o b e l o n g s to a p l a n a r d o m a i n on F , then b y c h a n g i n g the b e n d i n g f i e l d on ~ one c a n m a k e the d i s t a n c e r ( x , y) s t a t i o n a r y . O t h e r w i s e , we t u r n the p l a n e Q s u f f i c i e n t l y s l i g h t l y so t h a t it s h o u l d i n t e r s e c t a p l a n a r d o m a i n on F ' in which the p o i n t s x and y lie and s h o u l d c o n t a i n the p o i n t s x and y . In the s e c t i o n we g e t a s u r f a c e F " on w h i c h the p o i n t s x and y w i l l no l o n g e r b e l o n g to an (n - 2 ) - d i m e n sional planar domain. S i n c e F " i s r i g i d o u t s i d e p l a n a r d o m a i n s , the d i s t a n c e r ( x , y) w i l l be s t a t i o n a r y . The c a p w t o g e t h e r with the b a s e p l a n e f o r m a c l o s e d c o n v e x h y p e r s u r f a e e w h i c h w i l l be r i g i d o u t s i d e p l a n a r d o m a i n s . C o n s e q u e n t l y , the c a p w i s a l s o r i g i d o u t s i d e p l a n a r d o m a i n s . We s h a l l p r o v e T h e o r e m 12. L e t F1 and F2 b e c o n v e x s m o o t h i s o m e t r i c h y p e r s u r f a c e s ; l e t P l be a p o i n t of s t r i c t c o n v e x i t y of the h y p e r s u r f a c e F1, and P2 be t h e p o i n t on F 2 c o r r e s p o n d i n g to it u n d e r the isometry. We m a t c h the p o I n t s P l and P2 and d i r e c t i o n s in the t a n g e n t h y p e r p i a n e s at the p o i n t s P l and P2 w h i c h correspond under the isometry. We c o n s i d e r the h y p e r s u r f a c e d e f i n e d b y the r a d i u s v e c t o r l

r = ~ (r, + r2).

It, a s w a s p r o v e d e a r l i e r ,

w i l l be c o n v e x a n d the f i e l d T=rl--r2

w i l l be a b e n d i n g fieLd of it. We denote the m a t c h e d p o s i t i o n s of the p o i n t s P l and P2 b y P. We m o v e the s u p p o r t p l a n e at t h e p o i n t P paraLLel so t h a t it s h o u l d cut f r o m the s u r f a c e 9 a c a p w. The c a p w w i l l be r i g i d

1302

o u t s i d e p l a n a r d o m a i n s , i . e . , the f i e l d is t r i v i a l o u t s i d e p l a n a r d o m a i n s . o n l y a s l i n e a r c o m b i n a t i o n s of p l a n a r d o m a i n s on F 1 a n d F 2.

P l a n a r d o m a i n s on w can be o b t a i n e d

C o n s e q u e n t l y , s i n c e the d o m a i n s on F 1 and F 2 c o r r e s p o n d i n g to co c o i n c i d e o u t s i d e p l a n a r d o m a i n s , they must coincide completely. 2.

Local

Nonbendability

of

Convex

Surfaces

In t h i s s e c t i o n the l o c a l n o n b e n d a b i l i t y t h e o r e m s will be s t r e n g t h e n e d . T H E O R E M 14. A c o n v e x h y p e r s u r f a c e n o t c o n t a i n i n g p l a n a r d o m a i n s of d i m e n s i o n n - 1 i s r i g i d in a n e i g h b o r h o o d of e a c h p o i n t n o t l y i n g in a p l a n a r d o m a i n of d i m e n s i o n n - 2 and n - 3. Now if the h y p e r s u r f a c e c o n t a i n s a n Gq - 1 ) - d i m e n s i o n a l p l a n a r d o m a i n , then it is r i g i d in a n e i g h b o r h o o d of the p o i n t s i n d i c a t e d o u t side (n - 1 ) - d i m e n s i o n a l p l a n a r d o m a i n s . T H E O R E M 15. L e t F I and F 2 b e i s o m e t r i c s m o o t h c o n v e x h y p e r s u r f a c e s . L e t Pt b e a p o i n t on F 1 n o t b e l o n g i n g to p l a n a r d o m a i n s of d i m e n s i o n s n - 1, n - 2, n - 3, and P2 be the p o i n t o n F 2 which c o r r e s p o n d s u n d e r the i s o m e t r y to Pl- T h e n s u f f i c i e n t l y s m a l l n e i g h b o r h o o d s of the p o i n t s P t and 1 2 a r e c o n g r u e n t . P r o o f of T h e o r e m 14. L e t x 0 b e a p o i n t of the h y p e r s u r f a c e F n o t b e l o n g i n g to p l a n a r d o m a i n s of d i m e n s i o n s n - 1, n - 2, n - 3. If x0 is a point of s t r i c t c o n v e x i t y , t h e n , as p r o v e d in Sec. 1, the h y p e r s u r face in a n e i g h b o r h o o d of the point x 0 i s r i g i d o u t s i d e (n - 1 ) - d i m e n s i o n a l p l a n a r d o m a i n s . We s h a l l show t h a t it is r i g i d o u t s i d e (n - 1 ) - d i m e n s i o n a l p l a n a r d o m a i n s in a n e i g h b o r h o o d of the p o i n t x 0, if x 0 b e l o n g s to a p l a n a r d o m a i n of d i m e n s i o n not h i g h e r than n - 4. The p r o o f w i l l go by i n d u c t i o n . A t h r e e - d i m e n s i o n a l h y p e r s u r f a c e of f o u r - d i m e n s i o n a l s p a c e is r i g i d in the n e i g h b o r h o o d of a p o i n t of s t r i c t c o n v e x i t y (a z e r o - d i m e n s i o n a l p l a n a r d o m a i n ) o u t s i d e t h r e e - d i m e n s i o n a l planar domains. L e t us a s s u m e t h a t an (n - 2 ) - d i m e n s i o n a l h y p e r s u r f a c e of (n - 1 ) - d i m e n s i o n a l s p a c e is r i g i d in a n e i g h b o r h o o d of a p o i n t of a p l a n a r d o m a i n of d i m e n s i o n (n - 1) - k (k -> 4) o u t s i d e (n - 2 ) - d i m e n s i o n a l p l a n a r d o m a i n s , and we s h a l l show t h a t an (n - 1 ) - d i m e n s i o n a l h y p e r s u r f a c e of n - d i m e n s i o n a l s p a c e is r i g i d in a n e i g h b o r h o o d of a p o i n t of a p l a n a r d o m a i n of d i m e n s i o n n - k (k -> 4) o u t s i d e (~ - 1 ) , d i m e n s i o n a l p l a n a r domains. L e t the p o i n t x0 of the h y p e r s u r f a c e F b e l o n g to a p l a n a r d o m a i n L of d i m e n s i o n n - k (k -> 4). We d r a w t h r o u g h the p o i n t x0 a h y p e r p l a n e P , i n t e r s e c t i n g the p l a n a r d o m a i n L. It i n t e r s e c t s the h y p e r s u r f a c e F in s o m e (n - 2 ) - d i m e n s i o n a l c o n v e x s u r f a c e F t , while on F t the p o i n t x 0 will b e l o n g to a p l a n a r d o m a i n of d i m e n s i o n ba - 1) - k (k >- 4). By the i n d u c t i v e h y p o t h e s i s F~ will be r i g i d in the h y p e r p l a n e P o u t s i d e (n - 2 ) - d i mensional planar domains. We s h a l l show t h a t F 1 will b e r i g i d in P a l s o on (n - 2 ) - d i m e n s i o n a l p l a n a r d o m a i n s . L e t the p o i n t s x a n d y on F 1 n o t b e l o n g to an (n - 1 ) - d i m e n s i o n a l p l a n a r d o m a i n on F and at l e a s t one of t h e m b e l o n g to an ( n - 2 ) - d i m e n s i o n a l p l a n a r d o m a i n on F 1. We t u r n the h y p e r p l a n e P a r o u n d the s e g m e n t xy into the p o s i t i o n P~. If the r o t a t i o n is s u f f i c i e n t l y s m a l l , then Pt i n t e r s e c t s the p l a n a r d o m a i n L in a p l a n a r d o m a i n of d i m e n s i o n (n - 1) - k (k -> 4), and the h y p e r s u r f a c e F in s o m e (n - 2 ) - d i m e n s i o n a l s u r f a c e F 2. On F 2 the p o i n t s x and y will no l o n g e r b e l o n g to (n - 2 ) - d i m e n s i o n a l p l a n e s , s i n c e o t h e r w i s e they would have to b e l o n g to a n (n - 1 ) - d i m e n s i o n a l p l a n a r d o m a i n on F.

B y the i n d u c t i v e h y p o t h e s i s the s u r f a c e F 2 ~S r i g i d o u t s i d e (n - 2 ) - d i m e n s i o n a l p l a n a r d o m a i n s . C o n s e q u e n t l y , the d i s t a n c e b e t w e e n the p o i n t s x and y i s s t a t i o n a r y and this m e a n s that F 1 is r i g i d i n P also on (n - 2 ) - d i m e n s i o n a l p l a n a r d o m a i n s . T h u s , a l l s e c t i o n s of the h y p e r s u r f a c e F by h y p e r p l a n e s p a s s i n g t h r o u g h the p o i n t x 0 and i n t e r s e c t i n g L a r e r i g i d o u t s i d e p o i n t s on (n - 1 ) - d i m e n s i o n a l p l a n a r d o m a i n s . The s a m e thing is a l s o t r u e f o r a l l s e c t i o n s p a s s i n g t h r o u g h any p o i n t c l o s e to x 0 a n d i n t e r s e c t i n g L. But t h i s m e a n s that the h y p e r s u r f a c e F is r i g i d in a n e i g h b o r h o o d of the p o i n t x 0 o u t s i d e (n - 1 ) - d i m e n s i o n a l p l a n a r d o m a i n s . P r o o f of T h e o r e m 15. We m o v e the p o i n t s P l and P2 to m a t c h so that the t a n g e n t h y p e r p l a n e s at t h e s e p o i n t s s h o u l d c o i n c i d e a n d a l s o d i r e c t i o n s which c o r r e s p o n d u n d e r the i s o m e t r y . L e t r~ a n d r2 be the r a d i u s v e c t o r s of the h y p e r s u r f a c e s F 1 and F 2.

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We c o n s i d e r the h y p e r s u r f a c e ~ with r a d i u s , v e c t o r 1

r = ~ (~, + r2). As p r o v e d in See. 1, it w i l l be c o n v e x in a n e i g h b o r h o o d of the p o i n t P , w h e r e P is the c o m m o n p o s i t i o n of the p o i n t s P l and P2. The v e c t o r f i e l d T = r 1 - r2 will b e a b e n d i n g for ~ . Since the p o i n t P l on F I does n o t b e l o n g to a p l a n a r d o m a i n of d i m e n s i o n n - 1, n - 2, n - 3, on r it will n o t b e l o n g to a p l a n a r d o m a i n of d i m e n s i o n n - 1, n - 2, n - 3. By T h e o r e m 14 the h y p e r s u r f a e e ~ will be r i g i d in a neighborhood, of the point P o u t s i d e (n - 1 ) - d i m e n s i o n a l p l a n a r d o m a i n s , b u t (n - D - d i m e n s i o n a l p l a n a r d o m a i n s a r e o b t a i n e d o n ~ only in the c a s e when the d o m a i n s o n F1 a n d F 2 c o r r e s p o n d i n g to t h e m a r e a l s o (n - D - d i m e n s i o n a l and c o r r e s p o n d u n d e r the i s o m e t r y . C o n s e q u e n t l y , the field T w i l l be t r i v i a l o u t s i d e 61 - D - d i m e n s i o n a l p l a n a r d o m a i n s on 6 , a n d this m e a n s that the h y p e r s u r f a c e s F i and F 2 c o i n c i d e o u t s i d e (n - 1 ) - d i m e n s i o n a l p l a n a r d o m a i n s . B u t t h e n they c o i n c i d e a l s o on the (n - D - d i m e n s i o n a l p l a n a r d o m a i n s .

LITERATURE i. 2. 3. 4. 5. 6. 7. 8. 9. I0. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27.

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CITED

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