Ricci tensor on ${\rm RCD}^*(K,N)$ spaces

TL;DR

This paper develops an improved Bochner inequality for ${\rm RCD}^*(K,N)$ spaces and introduces a new definition of the $N$-dimensional Ricci tensor applicable to metric measure spaces.

Contribution

It provides a novel formulation of the Ricci tensor on metric measure spaces under the ${\rm RCD}^*(K,N)$ condition, advancing geometric analysis in non-smooth settings.

Findings

Enhanced Bochner inequality for ${\rm RCD}^*(K,N)$ spaces

New definition of $N$-dimensional Ricci tensor in metric measure spaces

Potential applications to geometric analysis and curvature bounds

Abstract

We obtain an improved Bochner inequality based on the curvature-dimension condition ${\rm RCD}^*(K,N)$ and propose a definition of $N$-dimensional Ricci tensor on metric measure spaces.