On the conjugate locus of pseudo-Riemannian manifolds

Abstract

Let exp m : T mM → M be the exponential map of a Riemannian manifold M at a point m ∈ M. Warner proved that in any neighbourhood of a conjugate point in T mM , the map exp m is not injective. Moreover, he described the exponential map in a suitable coordinate system in a neighbourhood of a regular conjugate point, these points build an open dense set in the conjugate locus. We will investigate in the pseudo-Riemannian case such subsets, where the results of Warner generalize. For the definition of these subsets of the conjugate locus we use a bilinear form on ker( T v exp m ), where v is a conjugate point, which will defined by the geodesic flow and the pseudo-Riemannian metric tensor.