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

Abstract

Let $ M^{n+1} $ ($ n \ge 2 $) be a simply-connected space form of sectional curvature $ -\kappa^2 $ for some $ \kappa \geq 0 $, and $ I $ an interval not containing $ [-\kappa,\kappa] $ 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 sphere $ S^n $. 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 $ \mathscr{F} $ of all such closed hypersurfaces is either empty or weakly homotopy equivalent to the group of orientation-preserving diffeomorphisms of $ S^n $. An equivalence assigns to each element of $ \mathscr{F} $ a suitable modification of its Gauss map. For $ M $ not simply-connected, $ \mathscr{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 $.