Geometric curvature bounds in Riemannian manifolds with boundary

Abstract

An Alexandrov upper bound on curvature for a Riemannian manifold with boundary is proved to be the same as an upper bound on sectional curvature of interior sections and of sections of the boundary which bend away from the interior. As corollaries those same sectional curvatures are related to estimates for convexity and conjugate radii; the Hadamard-Cartan theorem and Yau’s isoperimetric inequality for spaces with negative curvature are generalized.