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 sem i-Riemannian geometry at geodesics with two variable endpoints is reduced t o the case of a fixed final endpoint. Using this observation, we give an el ementary proof of the Morse index theorem for Riemannian geodesics with two variable endpoints, in the spirit of the original Morse proof. This approa ch reduces substantially the effort required in the proofs of the theorem g iven 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 arg ument works also in the case of timelike geodesics between two submanifolds of a Lorentzian manifold. For the extension to the lightlike Lorentzian ca se, just minor changes are required and one obtains easily a proof of the f ocal index theorem previously presented [J. Geom. Phys. 6(4), 657-670 (1989 )]. (C) 1999 American Institute of Physics. [S0022- 2488(99)01012-9].