A Morse complex for Lorentzian geodesics

We prove the Morse relations for all geodesics connecting two non-conjugate points on a class of globally hyperbolic Lorentzian manifolds. We overcome the difficulties coming from the fact that the Morse index of every geodesic is infinite, and from the lack of the Palais-Smale condition, by using the Morse complex approach. Introduction Let M be a smooth connected manifold without boundary of dimension n + 1, and let h be a Lorentzian structure on M : this means that h is a non-degenerate symmetric (0,2)-tensor on M having n positive eigenvalues and one negative eigenvalue (see [O’N83] and [BEE96] for foundational results on Lorentzian geometry). The Lorentzian structure h induces a unique Levi-Civita covariant derivative ∇ on M , and a geodesic on (M,h) is a curve γ : I → M whose velocity γ̇ is parallelely transported with respect to this covariant derivative. We are interested in the problem of classifying all geodesics connecting two fixed points z0, z1 in M . These geodesics are critical points of the energy functional