Morse functions and submanifolds of hyperbolic space
Toshihiko Ikawa · Rocky Mountain Journal of Mathematics · 1984
We study the hypersurface of the Hyperbolic space H Zm+ \ In // 2m+1 , there are Morse functions.If we assume that these Morse functions have index 0, m or 2m at all these critical points, then we can determine the hypersurface. Introduction.Let M be a differentiate manifold of class C°°.By a Morse function / of M, we mean a differentiate function on M having only non-degenerate critical points.In [5], Nomizu and Rodriguez showed the following result of a geometric nature analogous to Reeb's Theorem.If M (dimAf = n § 2) is a connected, complete Riemannian manifold isometrically immersed in R n+ P such that every Morse function of the form L p has index 0 or n at any of its critical points, then M is embedded as a Euclidean subspace or a Euclidean w-sphere.Here L p (x) = (d(x, p)) 2 , p e R n+ P, x e M and d is the Euclidean distance function (see also [4]).Cecil [1] characterized the metric spheres in hyperbolic space H m in terms of hyperbolic distance functions L p .In [2], Cecil and Ryan studied umbilic submanifolds in a hyperbolic space through the introduction of new classes of Morse functions, L n (directed distance from a hyperplane) and L h (directed distance from a horosphere).They proved the following theorem.THEOREM A. Let M n , (n ^ 2), be a connected, complete Riemannian manifold isometrically immersed in H m .Every Morse function of the form L p or L n has index 0 or n at all its critical points if and only if M n is embedded as a sphere, horosphere or equidistant hypersurface in a totally geodesic ffn+l c H mIn this paper, we shall study more general submanifolds in a hyperbolic space using Morse functions.For background material and notation, we refer the reader to [2].