A note on the Morse index theorem for geodesics between submanifolds in semi-Riemannian geometry

The computation of the index of the Hessian of the action functional in semi-Riemannian geometry at geodesics with two variable endpoints is reduced to the case of a fixed final endpoint. Using this observation, we give an elementary proof of the Morse index theorem for Riemannian geodesics with two variable endpoints, in the spirit of the original Morse proof. This approach reduces substantially the effort required in the proofs of the theorem given previously [Ann. Math. 73(1), 49–86 (1961); J. Diff. Geom 12, 567–581 (1977); Trans. Am. Math. Soc. 308(1), 341–348 (1988)]. Exactly the same argument works also in the case of timelike geodesics between two submanifolds of a Lorentzian manifold. For the extension to the lightlike Lorentzian case, just minor changes are required and one obtains easily a proof of the focal index theorem previously presented [J. Geom. Phys. 6(4), 657–670 (1989)].