A characterization of metric spheres in hyperbolic space by Morse theory
Thomas E. Cecil · Tohoku Mathematical Journal · 1974
Introduction.Let M n be a differentiable manifold of class C°°.By a Morse function / on M n , we mean a differentiable function / on M n 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 n having exactly 2 critical points, then M n is homeomorphic to an ?^-sphere, S n (see, for example, [3], p. 25).In a recent paper, [4], Nomizu and Rodriguez found a geometric characterization of a Euclidean ^-sphere S n aR n+p in terms of the critical point behavior of a certain class of functions L p , p e R n+P , on M n .In that case, if p e R n+P , x e M n , then L p (x) = (d(x, p))\ where d is the Euclidean distance function.Nomizu and Rodriguez proved that if M n {n ^ 2) is a connected, complete Riemannian manifold isometrically immersed in R n+P such that every Morse function of the form L p , p e R n+P , has index 0 or n at any of its critical points, then M n is embedded as a Euclidean subspace, R n , or a Euclidean ^-sphere, S n .This result includes the following: if M n is compact such that every Morse function of the form L p has exactly 2 critical points, then M n = S n .In this paper, we prove results analogous to those of Nomizu and Rodriguez for a submanifold M n of hyperbolic space, H n+P , the spaceform of constant sectional curvature -1.For p € H n+P , x e M n , we define the function L p (x) to be the distance in H n+P from p to x.We then define the concept of a focal point of (M n , x) and prove an Index Theorem for L p which states that the index of L p at a non-degenerate critical point x is equal to the number of focal points of {M n , x) on the geodesic in H n+P from x to p.In section 2, we prove that a metric sphere S n c H n+P can be characterized by the condition that every Morse function of the form L p , p e H n+P , has exactly 2 critical points.In section 3, we give an example which shows that a result analo-