Moments of Sectional Curvature

TL;DR

This paper computes the moments and distribution of sectional curvature as a random variable on compact Riemannian manifolds, providing explicit results for symmetric spaces and implications for geometric inequalities.

Contribution

It introduces a method to calculate moments of sectional curvature using local invariants and derives explicit formulas for symmetric spaces, linking curvature distribution to geometric inequalities.

Findings

Calculated moments of sectional curvature for symmetric spaces.

Derived distribution formulas for sectional curvature.

Proved a weak version of the Hitchin-Thorpe Inequality.

Abstract

The sectional curvature of a compact Riemannian manifold M can be seen as a random variable on the Grassmann bundle of 2-planes in TM endowed with the Fubini-Study volume density. In this article we calculate the moments of this random variable by integrating suitable local Riemannian invariants and discuss the distribution of the sectional curvature of Riemannian products. Moreover we calculate the moments and the distribution of the sectional curvature for all compact symmetric spaces of rank 1 explicitly and derive a formula for the moments of general symmetric spaces. Interpolating the explicit values for the moments obtained we prove a weak version of the Hitchin-Thorpe Inequality.