#### Document Type

Article

#### Publication Date

1974

#### Department

Mathematics

#### Abstract

Let *M*^{n} be a differentiable manifold of class C^{¥}. By a Morse function *f* on *M ^{n}*, we mean a differentiable function

*f*on

*M*having only non-degenerate critical points. A well-known topological result of Reeb states that if

^{n}*M*is compact and there is a Morse function

^{n}*f*on

*M*having exactly 2 critical points, then

^{n}*M*is homeomorphic to an

^{n}*n*-sphere,

*S*(see, for example, [3], p. 25).

^{n}In a recent paper, [4], Nomizu and Rodriguez found a geometric characterization of a Euclidean *n*-sphere *S ^{n Ì }R^{n+p}* in terms of the critical point behavior of a certain class of functions

*L*,

_{p}*p*Î

*R*, on

^{n+p}*M*. In that case, if

^{n}*p*Î

*R*Î

^{n+p},^{ }x*M*then

^{n},*L*=

_{p}(x)*(d(x, p))*where

^{2}*d*is the Euclidean distance function.

Nomizu and Rodriguez proved that if *M ^{n}* (

*n*³ 2) is a connected, complete Riemannian manifold isometrically immersed in

*R*such that every Morse function of the form

^{n+p}*L*,

_{p}*p*Î

*R*, has index 0 or

^{n+p}*n*at any of its critical points, then

*M*is embedded as a Euclidean subspace,

^{n}*R*, or a Euclidean

^{n}*n*-sphere,

*S*. This result includes the following: if

^{n}*M*is compact such that every Morse function of the form

^{n}*L*has exactly 2 critical points, then

_{p}*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*, the spaceform of constant sectional curvature —1.

^{n+p}#### Recommended Citation

Cecil, Thomas E. A characterization of metric spheres in hyperbolic space by Morse theory. Tohoku Math. J. 26 (1974), no. 3, 341-351.

## Comments

This is the publisher‘s version of the work. This publication appears in The College of the Holy Cross’ institutional repository by permission of the copyright owner for personal use, not for redistribution.

Originally published in Tohoku Math J. Volume 26, Number 3 (1974), 341-351.

DOI:10.2748/tmj/1178241128