On the Equivalence of Generalized Ricci Curvatures

TL;DR

This paper establishes the equivalence of various definitions of generalized Ricci curvature and characterizes conditions for the symmetry of the total generalized Ricci tensor.

Contribution

It provides a unifying proof of different generalized Ricci curvature notions and identifies when the total generalized Ricci tensor is symmetric.

Findings

Proved equivalence of multiple generalized Ricci curvature definitions

Characterized symmetry conditions for the total generalized Ricci tensor

Unified understanding of generalized Ricci curvature concepts

Abstract

We prove the equivalence between the several notions of generalized Ricci curvature found in the literature. As an application, we characterize when the total generalized Ricci tensor is symmetric.