  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 \colon 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 $ \big( \Gamma\backslash
\mathrm{SO}_{n+2} \big) $ 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.

Geometric Topology;Differential Geometry