A Gauss-Bonnet-Chern type obstruction for Killing vector fields on Lorentzian manifolds

Abstract

A new curvature obstruction to the existence of a timelike (resp. causal) Killing or homothetic vector field $X$ on an even-dimensional (odd-dimensional) Lorentzian manifold, in terms of its timelike (resp. null) sectional curvature is given. As a consequence for the compact case, the well-known Gauss-Bonnet-Chern obstruction to the existence of semi-Riemannian metrics is extended from non-zero constant sectional curvature to non-zero timelike sectional curvature on $X$.

v2: Remark 6 and one reference included. 12 pages, no figures