A lower bound for the curvature integral under an upper curvature bound

Abstract

We prove that the integral of scalar curvature over a Riemannian manifold is uniformly bounded below in terms of its dimension, upper bounds on sectional curvature and volume, and a lower bound on injectivity radius. This is an analogue of an earlier result of Petrunin for Riemannian manifolds with sectional curvature bounded below.

added Section 1.2, Example 3.5, Lemma 5.2, a simplified proof of Proposition 5.4, and many other details and explanations

References in corpus (3)

• Polyhedral approximations of Riemannian manifolds
• Conjectures and Open Questions on the Structure and Regularity of Spaces with Lower Ricci Curvature Bounds
• Curvature tensor of smoothable Alexandrov spaces