  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 $-1$. Then it
follows from the work of Besson-Courtois-Gallot that $\pi_1(S)$ can be
represented by a hypersurface $S'$ in $N$ with volume less than or equal to
that of $S$. We study the equality case: if $\pi_1(S)$ cannot be represented by
a hypersurface $S'$ in $N$ with volume strictly smaller than that of $S$, 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.
