  In this paper, we prove that a Riemannian $n$-manifold $M$ with sectional
curvature bounded above by $1$ that contains a minimal $2$-sphere of area
$4\pi$ which has index at least $n-2$ has constant sectional curvature $1$. The
proof uses the construction of ancient mean curvature flows that flow out of a
minimal submanifold. As a consequence we also prove a rigidity result for the
Simon-Smith minimal spheres.
