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

Let $M^{n+1}$ ($n\ge 2$) be a simply-connected space form of sectional curvature $-{\kappa }^2$ for some $\kappa \ge 0$, 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 $\mathcal{F}$ 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 $\mathcal{F}$ a suitable modification of its Gauss map. For $M$ not simply-connected, $\mathcal{F}$ 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 $M$.