Homotopical and topological rigidity of hypersurfaces of spherical space forms

Abstract

The first main result is a topological rigidity theorem for complete immersed hypersurfaces of spherical space forms which extends similar results due to do Carmo/Warner, Wang/Xia and Longa/Ripoll. Under certain sharp conditions on the principal curvatures of such a hypersurface $f:N^n\to M^{n+1}$ $(n\ge 2)$, it asserts that the universal cover of $N$ must be diffeomorphic to the $n$-sphere $S^n$, and provides an upper bound for the order of the fundamental group of $N$ in terms of that of $M$. In particular, if $M=S^{n+1}$, then $N$ is diffeomorphic to $S^n$ and either $f$ or its Gauss map is an embedding. Let $J\subset (0,\pi )$ be any interval of length less than $\frac{\pi }{2}$. The second main result constructs a weak homotopy equivalence between the space of all complete immersed hypersurfaces of $M$ with principal curvatures in $\cot (J)$ and the twisted product of $(\Gamma \backslash {\mathrm{SO}}_{n+2})$ and ${\mathrm{Diff}}_{+}(S^n)$ by ${\mathrm{SO}}_{n+1}$, where $\Gamma$ is the fundamental group of $M$ regarded as a subgroup of ${\mathrm{SO}}_{n+2}$. Relying on another rigidity criterion due to Wang/Xia, the third main result constructs a homotopy equivalence between the space of all complete immersed hypersurfaces of $S^{n+1}$ whose Gauss maps have image contained in a strictly convex ball and the same twisted product, with $\Gamma$ the trivial group.

arXiv comment18 pages, no figures, comments are welcome. Differs from previous version by minor improvements in the exposition and addition of a few references

References in corpus (3)

• On the components of spaces of curves on the 2-sphere with geodesic curvature in a prescribed interval
• The homotopy type of spaces of locally convex curves in the sphere
• On a class of immersions of spheres into space forms of nonpositive curvature