 
Summary: communications in
analysis and geometry
Volume 16, Number 2, 251282, 2008
Lorentz and semiRiemannian spaces with
Alexandrov curvature bounds
Stephanie B. Alexander and Richard L. Bishop
A semiRiemannian manifold is said to satisfy R K (or R K)
if spacelike sectional curvatures are K and timelike ones are K
(or the reverse). Such spaces are abundant, as warped product con
structions show; they include, in particular, big bang Robertson
Walker spaces. By stability, there are many nonwarped prod
uct examples. We prove the equivalence of this type of curvature
bound with local triangle comparisons on the signed lengths of
geodesics. Specifically, R K if and only if locally the signed
length of the geodesic between two points on any geodesic triangle
is at least that for the corresponding points of its model triangle
in the Riemannian, Lorentz or antiRiemannian plane of curva
ture K (and the reverse for R K). The proof is by compari
son of solutions of matrix Riccati equations for a modified shape
operator that is smoothly defined along reparametrized geodesics
