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communications in analysis and geometry

Summary: communications in
analysis and geometry
Volume 16, Number 2, 251­282, 2008
Lorentz and semi-Riemannian spaces with
Alexandrov curvature bounds
Stephanie B. Alexander and Richard L. Bishop
A semi-Riemannian 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 non-warped 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 anti-Riemannian 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


Source: Alexander, Stephanie - Department of Mathematics, University of Illinois at Urbana-Champaign


Collections: Mathematics