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On the Equivalence of Generalized Ricci Curvatures

Gil R. Cavalcanti, Jaime Pedregal, Roberto Rubio

TL;DR

The paper addresses the problem of unifying several generalized Ricci curvature notions in Courant algebroids by proving their equivalence and exploring when the total generalized Ricci tensor is symmetric. The authors analyze generalized connections, divergences, and torsion, and introduce the pair $ (\mathcal{G},\operatorname{div})$ and the total curvature to show that the Ricci tensors arising from the GF, JV, and SV formalisms coincide under suitable hypotheses. They demonstrate independence of the Ricci tensors from the particular metric connection when the divergence is fixed, and establish a symmetry criterion for the total Ricci tensor that ties to compatibility of $(\mathcal{G},\operatorname{div})$ and to generalized Ricci flow. The results provide a coherent framework for generalized geometry curvature notions and have implications for flow equations in string theory and supergravity contexts.

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.

On the Equivalence of Generalized Ricci Curvatures

TL;DR

The paper addresses the problem of unifying several generalized Ricci curvature notions in Courant algebroids by proving their equivalence and exploring when the total generalized Ricci tensor is symmetric. The authors analyze generalized connections, divergences, and torsion, and introduce the pair and the total curvature to show that the Ricci tensors arising from the GF, JV, and SV formalisms coincide under suitable hypotheses. They demonstrate independence of the Ricci tensors from the particular metric connection when the divergence is fixed, and establish a symmetry criterion for the total Ricci tensor that ties to compatibility of and to generalized Ricci flow. The results provide a coherent framework for generalized geometry curvature notions and have implications for flow equations in string theory and supergravity contexts.

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.
Paper Structure (4 sections, 6 theorems, 51 equations)

This paper contains 4 sections, 6 theorems, 51 equations.

Table of Contents

  1. Introduction
  2. Generalized Ricci curvatures of a connection
  3. Generalized Ricci curvature of a pair $(\mathcal{G},\operatorname{div})$
  4. Total curvatures

Key Result

Theorem 8

For $\mathcal{G}$ a generalized metric and $D$ a metric connection on $E$, given any $a_\pm,b_\pm\in\Gamma(V_\pm)$,

Theorems & Definitions (25)

  • Definition 1
  • Definition 2
  • Definition 3: garcia-fernandez-heterotic
  • Definition 4: jurco-vysoky
  • Definition 5: garcia-fernandez-spinors
  • Definition 6: jurco-vysoky
  • Definition 7: streets-strickland-constable-valach
  • Theorem 8
  • proof
  • Definition 9: alekseev-xu
  • ...and 15 more