Differentiable rigidity of hypersurface in space forms (Q2017739)

In the main theorem of the paper under review the author proves a rigidity theorem for three-dimensional compact hypersurfaces \(M\) in a four-dimensional space form. The assumptions are the positivity of the scalar curvature and an integral inequality for the squared length of the second fundamental form. A similar result is shown for four-dimensional hypersurfaces in a five-dimensional space form where the inequality involves also the norm of the Weyl tensor. The rigidity is only topological in these cases, i.e., \(M\) is shown to be diffeomorphic with a standard space.