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.