Homotopical and topological rigidity of hypersurfaces of spherical space forms

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 $f\text{\hspace{0.17em}⁣}:N^n\to M^{n+1}$ $( n\ge 2 )$,itassertsthattheuniversalcoverof$ 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 $isdiffeomorphicto$ {S}^n $andeither$ 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 $ \big( \Gamma\backslash $\mathrm{S}\mathrm{O}${n+2} \big) $and$ $\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}$+({S}^n) $by$ $\mathrm{S}\mathrm{O}${n+1} $,where$ \Gamma $isthefundamentalgroupof$ M $regardedasasubgroupof$ $\mathrm{S}\mathrm{O}${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 $\mathrm{\Gamma }$ the trivial group.