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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 Sn Ì Rn+p in terms of the critical point behavior of a certain class of functions Lp, p Î Rn+p, on Mn. In that case, if p Î Rn+p, x Î Mn, then Lp(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 Rn+p such that every Morse function of the form Lp, p Î Rn+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 Lp 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, Hn+p, the spaceform of constant sectional curvature —1.


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.




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