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, , p. 25).
In a recent paper, , 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.
Cecil, Thomas E. A characterization of metric spheres in hyperbolic space by Morse theory. Tohoku Math. J. 26 (1974), no. 3, 341-351.