A Characterization of Metric Spheres in Hyperbolic Space by Morse Theory

Authors

Thomas E. Cecil, College of the Holy CrossFollow

Document Type

Article

Publication Date

1974

Department

Mathematics

Abstract

Let Mⁿ be a differentiable manifold of class C^¥. By a Morse function f on Mⁿ, we mean a differentiable function f on Mⁿ having only non-degenerate critical points. A well-known topological result of Reeb states that if Mⁿ is compact and there is a Morse function f on Mⁿ having exactly 2 critical points, then Mⁿ is homeomorphic to an n-sphere, Sⁿ (see, for example, [3], p. 25).

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^n+p, on Mⁿ. In that case, if p Î R^n+p, x Î Mⁿ, then L_p(x) = (d(x, p))² where d is the Euclidean distance function.

Nomizu and Rodriguez proved that if Mⁿ (n ³ 2) is a connected, complete Riemannian manifold isometrically immersed in R^n+p such that every Morse function of the form L_p, p Î R^n+p , has index 0 or n at any of its critical points, then Mⁿ is embedded as a Euclidean subspace, Rⁿ, or a Euclidean n-sphere, Sⁿ. This result includes the following: if Mⁿ is compact such that every Morse function of the form L_p has exactly 2 critical points, then Mⁿ = Sⁿ.

In this paper, we prove results analogous to those of Nomizu and Rodriguez for a submanifold Mⁿ of hyperbolic space, H^n+p, the spaceform of constant sectional curvature —1.

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

Repository Citation

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