Reports of the Academy of Sciences of the USSR

1. Volume 134, No. 6

MATHEMATICS

I. A. SOKOLENKO

SPHERES AND GEODESICS IN RIEMANNIAN SPACES WITH A POLE

(Presented by Academician S. L. Sobolev on 25 V 1960)

We consider a Riemannian space of positive curvature \(R^m\) with a pole \(O\) \((^1)\). The locus of points of \(R^m\) at distance \(\rho\) from \(O\) will be called the sphere of radius \(\rho\) with center at \(O\) and denoted by \(S_\rho\). \(R^m \setminus S_\rho\) consists of two components; the bounded component will be called the ball with center \(O\) of radius \(\rho\) and denoted by \(\Pi_\rho\), and the unbounded component will be denoted by \(T_\rho\). The sphere \(S_\rho\) is called strictly convex if, for any points \(M_1, M_2 \in S_\rho\), every shortest path \(M_1M_2\) lies in \(\Pi_\rho\), except for its endpoints.

For spaces \(R^m\) satisfying the above conditions, the following theorems are proved.

Theorem 1. All spheres of \(R^m\) with center at the pole \(O\) are strictly convex.

We note that from the proof given below there follows a stronger assertion: every geodesic arc (not necessarily a shortest one) with endpoints on \(S_\rho\) lies in \(\Pi_\rho\), except for its endpoints.

The following propositions on geodesics in \(R^m\) follow from Theorem 1.

Theorem 2. In the space \(R^m\) there are no closed geodesics.

Theorem 3. If the curvature of \(R^m\) at every point and in every two-dimensional direction is not greater than \(H\), then every geodesic arc of length \(\pi/\sqrt{H}\) is a shortest path.

The assertion of Theorem 3 was proved by A. V. Pogorelov \((^2)\) for closed two-dimensional convex surfaces and was extended by Klingenberg \((^3)\) to compact spaces of positive curvature of even dimension.

Proof of Theorem 1. Let \(M_1, M_2 \in S_\rho\), but not all other points of the shortest path \(M_1M_2\) belong to \(\Pi_\rho\). Then the following cases are possible:

I. \(M_1M_2\) contains some arc \(N_1N_2 \subset \overline{T}_\rho\) (\(\overline{T}_\rho\) is the closure of \(T_\rho\)). Joining \(O\) to \(N_1\) and \(N_2\) by shortest paths, we obtain the triangle \(ON_1N_2\). The angles \(ON_1N_2\) and \(ON_2N_1\), obviously, are not less than \(\pi/2\), which contradicts the comparison theorem \((^1)\).

II. The shortest path \(M_1M_2 \subset \overline{\Pi}_\rho\), and \(M_1M_2 \cap S_\rho\) is a closed nowhere dense set on \(M_1M_2\), not reducible to the two points \(M_1, M_2\). Then there exist at least two distinct intervals complementary to this set. Choose on these intervals one point each, so that these points \(N_1, N_2\) lie on some sphere \(S_{\rho_1}\), \(\rho_1 < \rho\); the existence of such points follows easily from the continuity of distance. The arc \(N_1N_2 \subset M_1M_2\) is a shortest path and, according to what was proved in case I, cannot contain points of \(T_{\rho_1}\); consequently, \(N_1N_2 \subset \overline{\Pi}_{\rho_1} \subset \Pi_\rho\). This contradicts the definition of the points \(N_1, N_2\), and Theorem 1 is proved.

Proof of Theorem 2. Suppose that in \(R^m\) there exists a closed geodesic. Let \(\rho = \max \rho(O, P)\), where \(P \in g\), and \(\rho(O, P)\) is the dist—

distance in \(R^m\). Then \(g\subset \overline{\Pi}_{\rho}\) and \(g\cap S_{\rho}\ne 0\). By Theorem 1, \(g\cap S_{\rho}\) cannot contain any arc and, consequently, is a closed nowhere dense subset of \(g\). In the complementary set \(g\setminus (g\cap S_{\rho})\) one can choose points \(N_1,N_2\) with the same properties as in case II of the proof of Theorem 1, and so close that the smaller arc \(N_1N_2\subset g\) is shortest. This leads to a contradiction, just as in the proof of Theorem 1.

Proof of Theorem 3. Suppose that the assertion of Theorem 3 is false; then there exists a geodesic arc \(g_1\) of length \(\pi/\sqrt{H}\) which is not shortest. With the aid of Jacobi’s equations it is proved that no geodesic arc of length less than \(\pi/\sqrt{H}\) contains a pair of conjugate points.

Let \(g_2\) be a shortest arc joining the endpoints of \(g_1\); then \(g_1,g_2\) form a geodesic digon of perimeter less than \(2\pi/\sqrt{H}\). Take a closed ball \(\overline{\Pi}_{\rho}\) containing the constructed digon. Denote by \(\Omega\) the set of all geodesic digons of perimeter less than \(2\pi/\sqrt{H}\) lying in \(\overline{\Pi}_{\rho}\); evidently, \(\Omega\) is nonempty. The subsequent arguments are built on combining certain methods of A. V. Pogorelov \((^2)\) with our Theorems 1 and 2.

Let \(s_0\) be the lower bound of the lengths of the digons in \(\Omega\); let \(h_n\) be a sequence of digons in \(\Omega\) whose lengths converge to \(s_0\). One can choose \(h_n\) in such a way that their constituent geodesic arcs all have length less than \(\pi/\sqrt{H}\) and converge to some geodesic arcs \(g_3,g_4\). Since arcs of length less than \(\pi/\sqrt{H}\) contain no conjugate points, \(g_3\) and \(g_4\) do not coincide and form a certain digon \(g\in\Omega\). We shall show that \(g\) is a closed geodesic; then we arrive at a contradiction with Theorem 2, and Theorem 3 will be proved. If \(g\) is not a closed geodesic, then at one of the vertices of the digon \(g\), for example at \(P_1\), the arcs \(g_3,g_4\) form an angle smaller than \(\pi\). Denote by \(P_2\) the second vertex of the digon \(g\). By virtue of the absence on \(g_3\) and \(g_4\) of points conjugate to \(P_2\), one can surround \(g_3\) and \(g_4\) with central fields of geodesics \(G_3\) and \(G_4\) in such a way that some neighborhood \(U\) of the point \(P_1\) belongs to both fields. Take on \(g_3,g_4\), respectively, points \(P_3,P_4\), distinct from \(P_1\) and so close to \(P_1\) that the shortest arc \(P_3P_4\) belongs to \(U\). Then, by Theorem 1, the midpoint \(P_5\) of the shortest arc \(P_3P_4\) belongs to \(\Pi_{\rho}\). Since, moreover, \(P_5\in U\), in the fields \(G_3,G_4\) there are contained, respectively, geodesic arcs \(g'_3,g'_4\), joining \(P_2\) with \(P_5\); by Theorem 1, \(g'_3,g'_4\) lie in \(\Pi_{\rho}\). By Weierstrass’s theorem,

\[ \begin{aligned} &\text{the length of } g'_3 \text{ is less than the length of } P_2P_3+\text{the length of } P_3P_5,\\ &\text{the length of } g'_4 \text{ is less than the length of } P_2P_4+\text{the length of } P_4P_5. \end{aligned} \tag{1} \]

Since the angle at the point \(P_1\) is less than \(\pi\), we have

\[ \text{the length of } P_3P_4 \text{ is less than the length of } P_3P_1+\text{the length of } P_4P_1. \tag{2} \]

From (1), (2) it follows that the perimeter of the geodesic digon \(g'=(g'_3,g'_4)\) is less than \(s_0\); but \(g'\) belongs to \(\Omega\), which contradicts the definition of \(s_0\). The contradiction obtained proves Theorem 3.

The author expresses deep gratitude to Yu. B. Rumer and V. A. Toponogov for their attention to this work.

Novosibirsk State
Pedagogical Institute

Received
20 V 1960

CITED LITERATURE

\(^1\) I. A. Sokolenko, DAN, 134, No. 5 (1960).
\(^2\) A. V. Pogorelov, Matem. sborn., 18 (60), 181 (1946).
\(^3\) W. Klingenberg, Ann. of Math., 69, No. 3 (1959).