The Morse Index Theorem where the Ends are Submanifolds
Diane Kalish · Transactions of the American Mathematical Society · 1988
In this paper the Morse Index Theorem is proven in the case where submanifolds $P$ and $Q$ are at the endpoints of a geodesic, $\gamma$. At $\gamma$, the index of the Hessian of the energy function defined on paths joining $P$ and $Q$ is computed using $P$-focal points, and a calculation at the endpoint of $\gamma$, involving the second fundamental form of $Q$.