  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 $X$.
