UDC 513.731

MATHEMATICS

Corresponding Member of the Academy of Sciences of the USSR A. V. POGORELOV

ON A HOMEOMORPHISM OF CONVEX SURFACES PRESERVING THE VARIATIONS OF TURN OF CURVES

Let \(L(A_1A_2 \ldots A_n)\) be a polygonal line with vertices \(A_k\). The turn of the polygonal line \(L\) is the number

\[ \psi(L)=\sum_k(\pi-\alpha_k), \]

where \(\alpha_k\) is the angle between the segments of the polygonal line meeting at the vertex \(A_k\). The variation of turn of a curve is the exact upper bound of the turns of polygonal lines inscribed in this curve. Curves of bounded variation of turn were introduced by A. D. Aleksandrov and have been well studied. Some of their properties that are used below may be found in \((^1)\). The purpose of the present note is to prove the following theorem.

Theorem. A homeomorphic point correspondence between convex surfaces that preserves the variations of turn of curves is a similarity.

Let \(F\) and \(F'\) be two convex surfaces between which a homeomorphism \(f\) is established, preserving the variations of turn of curves. We shall first show that the homeomorphism \(f\) is a conformal correspondence. Take on the surface \(F\) an arbitrary point \(A\) and two directions from this point on the surface. In these directions, draw from the point \(A\) curves \(\gamma_1\) and \(\gamma_2\) of bounded variation of turn. Such curves are, for example, plane sections of the surface. On the surface \(F'\), the point \(A\) corresponds to the point \(A'\), and to the curves \(\gamma_1\) and \(\gamma_2\) correspond the curves \(\gamma_1'\) and \(\gamma_2'\), issuing from the point \(A'\). Since the curves \(\gamma_1'\) and \(\gamma_2'\) are of bounded variation of turn, they have definite directions (semitangents) at the point \(A'\).

Denote by \(\gamma\) the curve composed of the curves \(\gamma_1\) and \(\gamma_2\). The variation of its turn is

\[ \psi(\gamma)=\psi(\gamma_1)+\psi(\gamma_2)+(\pi-\alpha), \]

where \(\alpha\) is the angle between the curves \(\gamma_1\) and \(\gamma_2\) at the point \(A\). Similarly, for the curve \(\gamma'\), composed of the curves \(\gamma_1'\) and \(\gamma_2'\), we shall have

\[ \psi(\gamma')=\psi(\gamma_1')+\psi(\gamma_2')+(\pi-\alpha'), \]

where \(\alpha'\) is the angle between the curves \(\gamma_1'\) and \(\gamma_2'\) at the point \(A'\). By the hypothesis of the theorem, \(\psi(\gamma)=\psi(\gamma')\), \(\psi(\gamma_1)=\psi(\gamma_1')\), \(\psi(\gamma_2)=\psi(\gamma_2')\). Therefore \(\alpha=\alpha'\). Since the directions of the curves \(\gamma_1\) and \(\gamma_2\) at the point \(A\) were chosen arbitrarily, this means that the correspondence \(f\) is conformal.

Let \(\gamma\) be a closed plane curve on the surface \(F\). It is convex and therefore has variation of turn equal to \(2\pi\). Since the corresponding curve on the surface \(F'\) is also closed and also has variation of turn \(2\pi\), it is plane and convex. Indeed, if it is not plane, then a spatial quadrilateral can be inscribed in it, and its turn is certainly greater than \(2\pi\). If it is plane but not convex, then a nonconvex polygon can be inscribed in it, and its turn is greater than \(2\pi\).

In order not to burden the proof, let us now restrict ourselves to the case of smooth convex surfaces \(F\) and \(F'\). Let \(A\) be an arbitrary point of the surface \(F\), and \(A'\) the corresponding point of the surface \(F'\). Draw a plane separating the point \(A\) from the boundary of the surface \(F\). It divides the surface—

surface \(F\) into two parts. Let \(F_A\) be that one of these parts to which the point \(A\) belongs. Denote by \(F_A'\) the corresponding part of the surface \(F'\) under the homeomorphism \(f\). Obviously, the surface \(F_A'\) also has a plane boundary. In order to prove the similarity of the surfaces \(F\) and \(F'\), it is enough to prove the similarity of the surfaces \(F_A\) and \(F_A'\) for an arbitrarily chosen point \(A\). Therefore, without loss of generality, we shall assume that the original surfaces \(F\) and \(F'\) have plane boundary.

Take two arbitrary points \(A_1\) and \(A_2\) on the surface \(F\). Let \(A_1'\) and \(A_2'\) be the corresponding points on the surface \(F'\). Since the surface \(F\) is strictly convex and has a plane boundary, through the points \(A_1\) and \(A_2\) one can draw a plane section \(\gamma\) which does not intersect the boundary of the surface. The corresponding curve \(\gamma'\), passing through the points \(A_1'\) and \(A_2'\) on the surface \(F'\), will also be a plane section.

Let \(\sigma\) and \(\sigma'\) be the planes in which the curves \(\gamma\) and \(\gamma'\) lie; let \(\alpha_1\) and \(\alpha_2\) be the tangent planes to the surface \(F\) at the points \(A_1\) and \(A_2\); and let \(\alpha_1'\) and \(\alpha_2'\) be the tangent planes to the surface \(F'\) at the points \(A_1'\) and \(A_2'\). Construct a projective transformation of space which sends the points \(A_1\) and \(A_2\) to \(A_1'\) and \(A_2'\), the planes \(\alpha_1\) and \(\alpha_2\) to \(\alpha_1'\) and \(\alpha_2'\), and the plane \(\sigma\) to the plane \(\sigma'\). The last condition is attainable in view of the conformality of the correspondence in the plane pencils with centers \(A_1\) and \(A_1'\), \(A_2\) and \(A_2'\), which is determined by the intersection of the planes \(\sigma\) and \(\sigma'\) with the tangent planes \(\alpha_1\) and \(\alpha_1'\), \(\alpha_2\) and \(\alpha_2'\). This projective transformation has the following properties: 1) it preserves the angles between the lines of the pencil with center \(A_1\) in the plane \(\alpha_1\); 2) it preserves the angles formed by the lines of intersection of the planes \(\sigma\) with the planes \(\alpha_1\) and \(\alpha_2\). The first property follows from the conformality of the homeomorphism \(f\). The second property follows from the equality of the rotations of the curves \(\gamma\) and \(\gamma'\) between the points \(A_1\) and \(A_2\), \(A_1'\) and \(A_2'\).

It is proved that the projective transformation thus constructed is a similarity. Hence it follows that, if the surfaces \(F\) and \(F'\) are oriented in the same way, then, when they are brought into coincidence at the points \(A_1\) and \(A_1'\) and in the corresponding directions at these points, the rays \(A_1A_2\) and \(A_1'A_2'\) coincide, while the tangent planes of the surfaces at the points \(A_2\) and \(A_2'\) will be parallel.

Bring the surfaces \(F\) and \(F'\) into coincidence at the points \(A_1\) and \(A_1'\) and in the corresponding directions at these points. Take the point of coincidence as the origin of coordinates, and denote by \(r\) and \(r'\) the vectors of corresponding points of the surfaces \(F\) and \(F'\) in this position. Then

\[ r=\lambda r', \qquad dr=\mu\,dr'. \]

Differentiating the first equality and comparing it with the second, we conclude that \(d\lambda=0\), i.e. \(\lambda=\mathrm{const}\), and consequently the surfaces \(F\) and \(F'\) are similar.

Physical-Technical Institute of Low Temperatures
Academy of Sciences of the Ukrainian SSR
Kharkov

Received
5 XI 1969

CITED LITERATURE

\(^{1}\) A. V. Pogorelov, Extrinsic Geometry of Convex Surfaces, “Nauka,” 1969.