The Maslov–Morse index of indefinite metrics

We compute the Maslov–Morse index of geodesics on a manifold with indefinite metric. It is shown that the multiplicity of conjugate points in the sense of Maslov is equal to the signature of a quadratic form obtained by restricting the metric to the space of degeneracy for the projection of the tangent space onto the Lagrangian manifold, if this latter spans the principal axes of the metric.