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

TL;DR

This paper proves that certain hypersurfaces in nonpositively curved space forms are not only topologically but also homotopically rigid, with their space of such hypersurfaces being homotopy equivalent to the diffeomorphism group of the sphere.

Contribution

It establishes homotopical rigidity of immersed hypersurfaces with principal curvatures in a fixed interval in nonpositive curvature space forms, extending known topological rigidity results.

Findings

The space of such hypersurfaces is either empty or weakly homotopy equivalent to the diffeomorphism group of the sphere.

Homotopical rigidity is shown via a modification of the Gauss map.

For non-simply-connected space forms, the hypersurface space is a quotient of the universal cover case.

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 $.