MATHEMATICS

L. A. Shor

ON THE FLEXIBILITY OF CONVEX SURFACES WITH BOUNDARY

(Presented by Academician P. S. Aleksandrov, July 2, 1960)

In the present note we investigate the question of the flexibility of general convex surfaces homeomorphic to an open disk in the class of all convex surfaces*. For surfaces with boundary satisfying certain conditions, effectively verifiable necessary and sufficient conditions for rigidity are established:

1. Let \(F\) be a convex surface homeomorphic to an open disk, bounded by a simple closed curve \(\Gamma\) of bounded variation of rotation. Denote by \(\overline{F}\) the boundary of the convex hull of the surface \(F\). The complement \(\overline{F}-F\) of the surface \(F\) to \(\overline{F}\) is developed onto the plane, being thereby transformed into a domain \(Q\), homeomorphic to a closed disk, generally speaking multi-sheeted. We denote the boundary of the domain \(Q\) by \(L\). The gluing of the domain \(Q\) to the surface \(F\), which gives the surface \(\overline{F}\), is called trivial.

By virtue of A. D. Aleksandrov’s “gluing theorem” \((^{1})\) and A. V. Pogorelov’s theorem on the unique determination of a general closed convex surface \((^{2})\), the convex surface \(F\) is flexible if and only if there exists a deformation of the corresponding trivial gluing; that is, if the domain \(Q\) admits a continuous deformation preserving the conditions of gluing to the surface \(F\).

We shall agree to denote corresponding points of the curves \(L\) and \(\Gamma\) under the trivial gluing by the same letters. Let the conical points of the surface \(\overline{F}\) lying on the curve \(\Gamma\) be denoted by \(A_i\) \((i=1,2,\ldots)\), and call them vertices of type \(A\). Let \(\varphi_{i0}\) be the rotation of the curve \(L\) at the point \(A_i\) from the side of the domain \(Q\), and let \(\varphi_{i1}\) be the rotation of the curve \(\Gamma\) at the point \(A_i\) on the surface \(\overline{F}\) from the side of the surface \(F\).

Denote by \(\Phi\) the set of surfaces \(F\) satisfying the conditions: 1) the boundary \(L\) of the domain \(Q\) has no points of return; 2) each point \(A_i\) of type \(A\) has on the curve \(L\) some neighborhood consisting of two rectilinear segments \(l_{i1}\) and \(l_{i2}\), whose common endpoint is the point \(A\).

The subset of the set \(\Phi\) composed of those surfaces \(F\) to which single-sheeted domains \(Q\) correspond will be denoted by \(\Phi'\).

1. Let the convex surface \(F \in \Phi'\). Denote by \(\overline{Q}\) the convex hull of the domain \(Q\). Orient the domains \(Q\) and \(\overline{Q}\) by choosing, as the positive direction of traversal of their boundaries, that for which their interior parts lie on the left. Suppose that, when \(Q\) is traversed in the positive direction, the segment \(l_{i1}\) follows the segment \(l_{i2}\) \((i=1,2,\ldots)\).

Suppose that on the curve \(L\) there is only one point \(A_1\) of type \(A\), and that it is an essential vertex of the domain \(\overline{Q}\). A sufficiently small neighbor-

* A convex surface \(F\) is called flexible in the class of all convex surfaces (below, simply flexible) if it can be included in a continuous family of isometric convex surfaces \(F_t\) not equal to it \((t \in [0,1],\ F_0 \equiv F)\).

of this point on the boundary of $\overline Q$ consists of two rectilinear segments whose common endpoint is the point $A_1$. Denote these segments by $l'$ and $l''$, and suppose that, in traversing $\overline Q$ in the positive direction, the segment $l'$ follows $l''$. We shall agree to understand by the angle formed by one ray with another the angle through which the first ray must be rotated counterclockwise in order for it to coincide with the second. Denote the angle formed by the continuation of the segment $l''$ beyond the point $A_1$ with the segment $l_{11}$ by $\theta_1$, and the angle formed by the segment $l_{12}$ with the continuation of the segment $l'$ beyond the point $A_1$ by $\theta_2$.

Suppose that on the curve $L$ there is a finite or infinite number of vertices $A_i$ of type $A$; moreover, if there is only one such vertex, then one and only one supporting line $a$ to the domain $Q$ passes through the point $A_1$; if there is more than one such vertex, then all of them lie on one line $a$ supporting the domain $Q$. Choose on the line $a$ a positive direction so that the domain $Q$ lies to its right. Denote the angle formed by the negative direction of the line $a$ with the segment $l_{i1}$ by $\theta_{i1}$, and the angle formed by the segment $l_{i2}$ with the positive direction of the line $a$ by $\theta_{i2}$.

We single out from the set $\Phi'$ three classes of surfaces: $K'_1$, $K'_2$, $K'_3$.

To the class $K'_1$ we assign the surfaces $F \in \Phi'$ for which the total curvature (the area of the spherical image) is equal to $4\pi$.

To the class $K'_2$ we assign the surfaces $F \in \Phi'$ satisfying the following conditions: 1) the curvature of the curve $\Gamma$ on the surface $\overline F$ is concentrated at one point $A_1$ of type $A$; 2) the point $A_1$ is an essential vertex of the domain $\overline Q$; 3) at the vertex $A_1$ the inequalities
\[ \varphi_{11}<\varphi_{10}, \qquad \tfrac12 \min(\varphi_{10}+\varphi_{11},\pi)<\min(\theta_1,\theta_2). \]

To the class $K'_3$ we assign the surfaces $F \in \Phi'$ satisfying the conditions: 1) the curvature of the curve $\Gamma$ on the surface $\overline F$ is concentrated at two points $A_1$ and $A_2$ of type $A$; 2) the points $A_1$ and $A_2$ of the curve $L$ lie on one line $a$—a supporting line of the domain $Q$, and on the line $a$ there are no other points of the curve $L$; 3) at the vertices $A_1$ and $A_2$ the inequalities
\[ \tfrac12(\varphi_{i0}+\varphi_{i1})<\min(\theta_{i1},\theta_{i2}), \qquad i=1,2. \]

Theorem 1. A convex surface $F\in\Phi'$ is non-bendable if and only if it belongs to one of the classes $K'_1$, $K'_2$, $K'_3$.

1. Now let a convex surface $F \in \Phi$. In this case the domain $Q$, which completes the surface $F$ to the surface $\overline F$, may turn out to be many-sheeted. Let $M$ be some set of points in the domain $Q$. The set of points of the plane over which the set $M$ is situated will be denoted by $\widetilde M$. The convex hull of the domain $\widetilde Q$ will be denoted by $\overline Q$. If the many-sheeted domain $Q$ is such that all points $\widetilde A_i$ lie on the boundary of $\overline Q$ and none of them is a multiple point of the curve $\widetilde L$, then for it all definitions of Sec. 2 can be repeated, including the definitions of the angles $\theta_1$, $\theta_2$, $\theta_{i1}$, $\theta_{i2}$.

We single out from the set $\Phi$ three classes $K_1$, $K_2$, $K_3$.

To the class $K_1$ we assign the surfaces $F \in \Phi$ whose total curvature is equal to $4\pi$.

To the class $K_2$ we assign the surfaces $F \in \Phi$ satisfying the conditions: 1) the curvature of the curve $\Gamma$ on the surface $\overline F$ is concentrated at one point $A_1$ of type $A$; 2) the point $\widetilde A_1$ is not a multiple point of the curve $\widetilde L$; 3) the point $\widetilde A_1$ is an essential vertex of $\overline Q$; 4) at the vertex $A_1$ the inequalities
\[ \varphi_{11}<\varphi_{10}, \qquad \tfrac12 \min(\varphi_{10}+\varphi_{11},\pi)<\min(\theta_1,\theta_2). \]

To class \(K_3\) we assign surfaces \(F \in \Phi\) satisfying the following conditions: 1) the curvature of the curve \(\Gamma\) on the surface \(\overline F\) is concentrated at a finite number \(n \geqslant 1\) of points \(A_i\) \((i=1,2,\ldots,n)\) of type \(A\); 2) none of the points \(\widetilde A_i\) is a multiple point of the curve \(\widetilde L\); 3) for \(n=1\), through the vertex \(A_1\) of the domain \(Q\) there passes one and only one supporting line \(a\), and, whatever the points \(B\) and \(C\) of the curve \(L\) lying on the line \(a\) on different sides of the point \(A_1\) may be, the orientations of the domains \(Q\) and \(\overline Q\), determined by one and the same order of traversal of the points \(A_1,B\), and \(C\), are opposite; for \(n>1\), all vertices \(A_i\) of the domain \(Q\) lie on one of its supporting lines \(a\); on the line \(a\) there do not exist three distinct points of the curve \(L\), among them at least two of type \(A\), such that the orientations of the domains \(Q\) and \(\overline Q\), determined by one and the same order of traversal of these points, coincide; 4) at each of the vertices \(A_i\) the inequality holds

\[ {}^{1}\!/_{2}(\varphi_{i0}+\varphi_{i1})<\min(\theta_{i1},\theta_{i2}),\qquad i=1,2,\ldots,n. \]

Theorem 2. A convex surface \(F \in \Phi\) is rigid if and only if it belongs to one of the classes \(K_1,K_2,K_3\).

Obviously, Theorem 1 is a particular case of Theorem 2. From Theorem 2 and the author’s work \((^3)\) it follows that the necessary and sufficient conditions for rigidity of convex polyhedra homeomorphic to the open disk and bounded by a simple closed polygonal line are the same in the class of all convex polyhedra and in the class of all convex surfaces.

The proof of the necessity of the condition of Theorem 2 is carried out by means of an effective construction of a deformation of the trivial gluing. To the nine rigidity conditions that hold for polyhedra (see \((^3)\)) there are added two more: 1) on the surface \(\overline F\) the curvature of the curve \(\Gamma\), with the exception of points of type \(A\), is not equal to zero; 2) the number of points of type \(A\) of the surface \(\overline F\) is infinite.

Proof of sufficiency. The rigidity of surfaces \(F \in K_1\) is obvious. The rigidity of surfaces \(F \in K_2+K_3\) is proved in the following way. First it is established that if a surface \(F \in K_2+K_3\), then there is no such deformation of the corresponding trivial gluing under which the turns of the curve \(L\) change at a finite number of points. This is proved separately for surfaces of the classes \(K_2\) and \(K_3\), and is analogous to the proof of the corresponding assertions for polyhedra in \((^3)\). Then, by means of a limiting transition, it is established that for such surfaces a deformation of the trivial gluing is altogether impossible.

In the course of the proof of Theorem 2 the following theorem is established.

Theorem 3. A convex surface \(F \in \Phi\) is a limit of convex surfaces nontrivially isometric to it if and only if it is flexible.

Voroshilov Mining and Metallurgical Institute

Received
10 VI 1960

REFERENCES

\(^{1}\) A. D. Alexandrov, Intrinsic Geometry of Convex Surfaces, Moscow–Leningrad, 1948. \(^{2}\) A. V. Pogorelov, Unique Determination of General Convex Surfaces, Kiev, 1952. \(^{3}\) L. A. Shor, Mat. sbornik, 45 (87), No. 4, 471 (1958).