Rigidity of totally geodesic hypersurfaces in negative curvature

Abstract

Let $M$ be a closed hyperbolic manifold containing a totally geodesic hypersurface $S$, and let $N$ be a closed Riemannian manifold homotopy equivalent to $M$ with sectional curvature bounded above by $-1$. We study the following question: if ${\pi }_1(S)$ can be represented by a totally geodesic hyperbolic hypersurface in $N$, then must $N$ be isometric to $M$? We show that many such $S$ are rigid in the sense that the answer to this question is positive. On the other hand, we construct examples of $S$ for which the answer is negative.