Reports of the Academy of Sciences of the USSR

1. Volume 115, No. 4

MATHEMATICS

V. A. TOPONOGOV

THE CONVEXITY PROPERTY OF RIEMANNIAN SPACES OF POSITIVE CURVATURE

(Presented by Academician P. S. Aleksandrov on III 5, 1957)

For Riemannian spaces \(R^n\) of positive (negative) curvature the following property is known: the angles of every sufficiently small triangle made of shortest arcs are not less (not greater) than the corresponding angles of a plane triangle with sides of the same length (see, for example, \((^9)\), p. 209). A. D. Aleksandrov proved that the same assertion is true for arbitrary triangles in two-dimensional manifolds, without assumptions of smoothness \((^1)\). In the \(n\)-dimensional case the corresponding result “in the large” for \(R^n\) of negative curvature was proved by É. Cartan \((^3)\). H. Busemann generalized Cartan’s results to the so-called \(G\)-spaces of negative curvature \((^2)\). In the present article the following theorem is proved:

Theorem. Let \(R^n\) be a twice continuously differentiable Riemannian manifold with a complete metric of nonpositive curvature. Then every triangle composed of shortest arcs in \(R_n\) has angles not less than the corresponding angles of a plane triangle with sides of the same length.

The theorem is proved in the equivalent form of A. D. Aleksandrov’s “convexity condition” \((^1)\), p. 109).

Let us first assume that \(R^n\) has an analytic metric of strictly positive curvature. We shall denote by \((ABC)'\) the plane triangle with sides equal to the sides of the triangle \(ABC\), formed by shortest arcs in \(R^n\).

Lemma 1. Let shortest arcs \(AB_0\), \(AB_n\) \((n=1,2,\ldots)\), and \(B_0C\) be given, \(B_n \in B_0C\), \(B_n \ne B_0\). If \(AB_n\) converge to \(AB_0\), and \(B_0\) is not a point conjugate to \(A\) on \(AB_0\), then there exists an \(N\) such that for \(n \ge N\) the angles \(AB_0B_n\) and \(AB_nB_0\) are not less than the corresponding angles of the triangle \((AB_0B_n)'\).

Proof. Introduce on \(AB_0\) a parameter \(t\) \((0 \le t \le 1)\), proportional to length: \(Y=Y(t)\), \(A=Y(1)\). Let \(\xi^i(0)\) be the unit tangent vector to \(B_0C\) at the point \(B_0\); \(\xi^i(t)\) the vector obtained from \(\xi^i(0)\) by parallel displacement along \(B_0A\). Passing through each point \(Y(t)\) a geodesic in the direction \(\xi^i(t)\), we obtain a surface \(\Gamma\). It is not difficult to calculate that the curvature of \(\Gamma\) at the points of \(AB_0\) is equal to the curvature of \(R^n\) in the two-dimensional direction tangent to \(\Gamma\); therefore a sufficiently narrow strip \(\Gamma_0 \subset \Gamma\) around \(AB_0\) has positive curvature. It can be shown that, starting with some \(n\), shortest arcs \(\overline{AB_n}\), converging to \(AB_0\), exist on \(\Gamma_0\). According to A. D. Aleksandrov’s theorem for the two-dimensional case, the angle \(B_0\) of the triangle \(\overline{AB_0B_n}\) is not less than the angle \(B'_0\) of the triangle \((B_0\overline{AB_n})'\) and, consequently, is not less than the angle \(B'_0\) of the triangle \((AB_0B_n)'\), since \(\overline{AB_n} \ge AB_n\). The assertion concerning the angles \(B_n\) is proved analogously.

Lemma 2. Let the shortest arcs \(AB, CX_n, CB\) be given, with \(C \in AB\), \(X_n \in AB\), \(X_n \ne B\). Denote the angle \(AX_nC\) by \(\xi_n\), and the angle between \(AB\) and \(BC\) by \(\alpha\). If \(X_n \to B\), then there exists \(\lim\limits_{n\to\infty}\xi_n=\bar\alpha\), and \(\bar\alpha \le \alpha\).

Proof. Obviously, it is enough to prove that if the shortest arcs \(CX_n\) converge to the shortest arc \(CB\), then \(\bar\alpha \le \alpha\). Suppose that \(\bar\alpha > \alpha\). Then, starting with some \(n\), on \(CX_n\) there will be a point \(\bar D_n\) such that \(\bar D_n \to B\), and the angle between the shortest arcs \(\bar D_nB\) and \(BA\) is equal to \(\alpha\). We may restrict ourselves to the case when \(\alpha \ne 0\), \(\alpha \ne \pi\). Take on \(BC\) a point \(D_n\) such that \(D_nB=\bar D_nB\); it is easy to see that \(\bar D_nX_n=D_nX_n+o(a_n)\), where \(a_n=\bar D_nB\). Comparing the lengths \(\bar CD_nB\) and \(CB\), respectively \(CX_n\) and \(CD_nX_n\), we obtain \(CD_n<\bar CD_n\le CD_n+o(a_n)\), whence \(\bar CD_n=CD_n+o(a_n)\). Take on \(\bar CD_n\) a point \(E_n\) such that \(\bar D_nE_n=\bar D_nB\). It is not hard to prove that

\[ E_nB=2\bar D_nB\sin{\frac{\psi_n}{2}}+o(a_n), \]

where \(\psi_n\) is the angle between \(CD_n\) and \(\bar D_nB\). Hence

\[ CE_nB=\bar CD_n+\bar D_nB-2\bar D_nB+E_nB =CD_n+D_nB+2\bar D_nB\left(1-\sin{\frac{\psi_n}{2}}\right)+ \]

\[ +o(a_n) =CB-2D_nB\left(1-\sin{\frac{\psi_n}{2}}\right)+o(a_n). \]

Since \(\lim\limits_{n\to\infty}\psi_n=\pi-(\bar\alpha-\alpha)<\pi\), for sufficiently large \(n\) we have \(CE_nB<CB\); but this is impossible, since \(CB\) is a shortest arc.

The proof of the theorem can now be carried out according to the plan of the proofs of A. D. Aleksandrov ((\(^{1}\)), pp. 100–103). Let \(OP\) and \(OQ\) be shortest arcs; \(Y\) a fixed interior point of \(OP\); \(X\) a variable point of \(OQ\); \(OX=x\), \(OY=y\); \(\gamma(x,y)\) the angle at \(O'\) of the triangle \((OXY)'\); \(\alpha(x)\) the angle between any shortest arc \(YX\) and \(XO\). It is enough to prove that \(\gamma(x,y)\) is a nonincreasing function of \(x\). Using the analyticity of the metric of \(R^n\), one can show that by an arbitrarily small shift \(OQ\) may be replaced by a shortest arc \(\overline{OQ}\) having the following property: on \(\overline{OQ}\) there exists only a finite number \(M\) of points \(X\) conjugate to \(Y\) on some one of the shortest arcs \(YX\). We shall assume that \(OQ\) already has this property.

Two assertions are proved simultaneously:

1. \(\gamma(x,y)\) is a nonincreasing function of \(x\).
2. \(\alpha(x)\) is not smaller than the corresponding angle of the triangle \((OXY)'\), whatever the shortest arc \(XY\).

By Lemma 1, assertions 1 and 2 are true for sufficiently small \(x\). Let \(x_0\) be the least upper bound of those \(x\) for which 1 and 2 are true; it is required to prove that \(x_0=OQ\). Suppose, on the contrary, that \(x_0<OQ\). As \(X\to X_0\), \(x<x_0\), the assumptions of Lemma 2 are fulfilled; consequently, 1 and 2 are true also for \(X_0\). If one assumes that \(X_0\notin M\), then as \(X\to X_0\), \(x>x_0\), the conditions of Lemma 1 are fulfilled; considering the triangles \(OYX_0\) and \(YX_0X_1\) (\(x_1>x_0\)) and applying the “lemma on convex quadrilaterals” ((\(^{1}\)), p. 100), we obtain that 1 and 2 are true also for \(X_1\), which is impossible. Hence \(X_0\in M\). Now take any \(x_1>x_0\) such that \((X_0X_1)_1\cap M=0\). Applying the preceding considerations to the shortest arcs \(X_1Y\), \(X_1X_0\) (instead of \(OP\), \(OQ\)), we become convinced that 1 and 2 are valid for all \(X\in X_1X_0\), including \(X_0\). Using again the lemma on convex quadrilaterals, it is easy to see that 1 and 2 are true also for \(X_1\), which contradicts the definition of \(x_0\). Thus, \(x_0=OQ\), as was required to prove.

Approximating a twice continuously differentiable metric of strictly positive curvature by an analytic metric, it is easy to remove the requirement that the metric be analytic.

Finally, replacing the comparison plane triangles by triangles of the Lobachevsky plane (cf. (\(^{1}\)), p. 345), one can extend the theorem to the case of nonnegative curvature. The theorem is also generalized to spaces whose curvature is bounded below. Let us also note that the limit-

transition makes it possible to carry the theorem over to arbitrary complete convex surfaces in \(n\)-dimensional Euclidean space. Hence it follows:

Theorem. On an \((n-1)\)-dimensional complete convex surface (without smoothness assumptions) in \(n\)-dimensional Euclidean space, a shortest curve is uniquely determined by any sufficiently small arc of it.

This theorem is an \(n\)-dimensional generalization of A. D. Aleksandrov’s “theorem on the non-overlapping of shortest curves” ([1], p. 109).

In conclusion I express my deep gratitude to A. I. Fet for posing the problem and for his constant help in the work.

Novosibirsk Electrotechnical
Institute of Communications

Received
27 II 1957

REFERENCES

\(^{1}\) A. D. Aleksandrov, The Intrinsic Geometry of Convex Surfaces, Moscow–Leningrad, 1948.
\(^{2}\) H. Busemann, Acta Math., 80, 259 (1948).
\(^{3}\) É. Cartan, Geometry of Riemannian Spaces, Moscow–Leningrad, 1936.