 
Summary: LORENTZ AND SEMIRIEMANNIAN SPACES WITH
ALEXANDROV CURVATURE BOUNDS
STEPHANIE B. ALEXANDER AND RICHARD L. BISHOP
Abstract. 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 RobertsonWalker
spaces. By stability, there are many nonwarped product examples. We
prove the equivalence of this type of curvature bound with local triangle
comparisons on the signed lengths of geodesics. Specically, 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 curvature K (and the reverse for R K). The proof is by comparison
of solutions of matrix Riccati equations for a modied shape operator
that is smoothly dened along reparametrized geodesics (including null
geodesics) radiating from a point. Also proved are semiRiemannian
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
