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LORENTZ AND SEMI-RIEMANNIAN SPACES WITH ALEXANDROV CURVATURE BOUNDS
 

Summary: LORENTZ AND SEMI-RIEMANNIAN SPACES WITH
ALEXANDROV CURVATURE BOUNDS
STEPHANIE B. ALEXANDER AND RICHARD L. BISHOP
Abstract. 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 product examples. We
prove the equivalence of this type of curvature bound with local triangle
comparisons on the signed lengths of geodesics. Speci cally, 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 curvature K (and the reverse for R  K). The proof is by comparison
of solutions of matrix Riccati equations for a modi ed shape operator
that is smoothly de ned along reparametrized geodesics (including null
geodesics) radiating from a point. Also proved are semi-Riemannian
analogues to the three basic Alexandrov triangle lemmas, namely, the
realizability, hinge and straightening lemmas. These analogues are intu-
itively surprising, both in one of the quantities considered, and also in the

  

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

 

Collections: Mathematics