On a class of immersions of spheres into space forms of nonpositive curvature

Let Mn+1 (n = 2) be a simply-connected space form of sectional curvature -.2 for some. = 0, and I an interval not containing [-.,.] in its interior. It is known that the domain of a closed immersed hypersurface of M whose principal curvatures lie in I must be diffeomorphic to the n-sphere Sn. These hypersurfaces are thus topologically rigid. The purpose of this paper is to show that they are also homotopically rigid. More precisely, for fixed I, the space F of all such hypersurfaces is either empty or weakly homotopy equivalent to the group of orientation-preserving diffeomorphisms of Sn. An equivalence assigns to each element of F a suitable modification of its Gauss map. For M not simply-connected, F is the quotient of the corresponding space of hypersurfaces of the universal cover of M by a natural free proper action of the fundamental group of M.