We study the singularities of the exponential map in semi Riemannian locally
symmetric manifolds. Conjugate points along geodesics depend only on real
negative eigenvalues of the curvature tensor, and their contribution to the
Maslov index of the geodesic is computed explicitly. We prove that degeneracy
of conjugate points, which is a phenomenon that can only occur in
semi-Riemannian geometry, is caused in the locally symmetric case by the lack
of diagonalizability of the curvature tensor. The case of Lie groups endowed
with a bi-invariant metric is studied in some detail, and conditions are given
for the lack of local injectivity of the exponential map around its
singularities.Comment: LaTeX2e, 27 page