The canonical generalised Levi-Civita connection and its curvature

TL;DR

This paper introduces a canonical generalised Levi-Civita connection on Courant algebroids, providing explicit curvature formulas and invariants that unify and extend classical geometric concepts in generalised geometry.

Contribution

It constructs a unique, canonical generalised Levi-Civita connection for given data, resolving non-uniqueness issues and deriving explicit curvature decompositions in generalised geometry.

Findings

Decomposition of the generalised Riemann curvature tensor in classical terms

Formulas for generalised Ricci and scalar curvature invariants

Introduction of two new generalised scalar curvature invariants

Abstract

Given a (semi-Riemannian) generalised metric $\mathcal G$ and a divergence operator $\mathrm{div}$ on an exact Courant algebroid $E$, we geometrically construct a canonical generalised Levi-Civita connection $D^{\mathcal G, \mathrm{div}}$ for these data. In this way we provide a resolution of the problem of non-uniqueness of generalised Levi-Civita connections. Since the generalised Riemann tensor of $D^{\mathcal G, \mathrm{div}}$ is an invariant of the pair $(\mathcal G, \mathrm{div})$, we no longer need to discard curvature components which depend on the choice of the generalised connection. As a main result we decompose the generalised Riemann curvature tensor of $D^{\mathcal G, \mathrm{div}}$ in terms of classical (non-generalised) geometric data. Based on this set of master formulas we derive a comprehensive curvature tool-kit for applications in generalised geometry. This includes decompositions for the full generalised Ricci tensor, the generalised Ricci tensor, and three generalised scalar-valued curvature invariants, two of which are new.