The first main result is a topological rigidity theorem for complete immersed hypersurfaces of spherical space forms from which similar theorems due to Wang/Xia and Longa/Ripoll can be derived. Under certain sharp conditions on the principal curvatures of such a hypersurface f: Nn → Mn+1 (n ≥ 2), it asserts that the universal cover of N must be diffeomorphic to the n-sphere \(\mathbb{S}^{n}\), and provides an upper bound for the order of the fundamental group of N in terms of that of M. In particular, if \(M=\mathbb{S}^{n+1}\), then N is diffeomorphic to \(\mathbb{S}^{n}\) and either f or its Gauss map is an embedding.Let j ⊂ (0, π) 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 (Γ\SOn+2)and \(\mathrm{Diff}_{+}(\mathbb{S}^{n})\) by SOn+1, where Γ is the fundamental group of M regarded as a subgroup of SOn+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 \(\mathbb{S}^{n+1}\) whose Gauss maps have image contained in a strictly convex ball and the same twisted product, with Γ the trivial group.