A characterization of metric spheres in hyperbolic space by Morse theory

Abstract

0. Introduction. Let M be a differentiable manifold of class C°°. By a Morse function / on M, we mean a differentiable function / on M having only non-degenerate critical points. A well-known topological result of Reeb states that if M* is compact and there is a Morse function / on M having exactly 2 critical points, then M is homeomorphic to an ?^-sphere, S (see, for example, [3], p. 25). In a recent paper, [4], Nomizu and Rodriguez found a geometric characterization of a Euclidean ^-sphere SaR in terms of the critical point behavior of a certain class of functions Lp, p e R , on M. In that case, if p e R, x e M, then Lp(x) = (d(x, p)) where d is the Euclidean distance function. Nomizu and Rodriguez proved that if M {n ^ 2) is a connected, complete Riemannian manifold isometrically immersed in R such that every Morse function of the form Lp, p e R , has index 0 or n at any of its critical points, then M is embedded as a Euclidean subspace, R, or a Euclidean ^-sphere, S. This result includes the following: if M is compact such that every Morse function of the form Lp has exactly 2 critical points, then M = S. In this paper, we prove results analogous to those of Nomizu and Rodriguez for a submanifold M of hyperbolic space, H, the spaceform of constant sectional curvature —1. For p € H, x e M, we define the function Lp(x) to be the distance in H from p to x. We then define the concept of a focal point of (M, x) and prove an Index Theorem for Lp which states that the index of Lp at a non-degenerate critical point x is equal to the number of focal points of {M, x) on the geodesic in H from x to p. In section 2, we prove that a metric sphere S c H can be characterized by the condition that every Morse function of the form Lp, p e H, has exactly 2 critical points. In section 3, we give an example which shows that a result analo-