An Alternative Proof for the Upper Bound of Curvature Integral on Manifolds with Lower Sectional Curvature Bound

Abstract

Petrunin proved that the integral of scalar curvature in a unit ball is bounded from above in terms of the dimension of the manifold and the lower bound of the sectional curvature. In this paper, we give an alternative proof for this result. The main difference between this proof and Petrunin’s original proof is that we construct a stratified finite covering and apply it directly to the given manifold, rather than arguing by contradiction for a sequence of manifolds, which satisfy some technical lifting properties.

Keywords

Scalar curvature / sectional curvature / L¹-norm / Alexandrov space / stratification / covering