Conjugate points and Maslov index in locally symmetric semi-Riemannian manifolds

Details Conjugate points and Maslov index in locally symmetric semi-Riemannian manifolds Conjugate points and Maslov index in locally symmetric semi-Riemannian manifolds

2005-05-09

arxiv.org/abs/math/0505160v1

Mathematics

Differential Geometry

LaTeX2e, 27 pages

Scientific paper

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.

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