Differential calculus for generalized geometry and geometric Lax flows

TL;DR

This paper develops differential calculus tools within generalized geometry, extending classical notions like curvature and Ricci flow to the generalized tangent bundle, and introduces geometric Lax flows for these structures.

Contribution

It introduces a framework for differential calculus on generalized tangent bundles, extending classical geometric concepts and connecting Ricci flows to geometric Lax flows.

Findings

Extended curvature tensor for generalized connections.

Defined generalized Ricci flow and Ricci soliton concepts.

Linked Ricci flows to geometric Lax flows.

Abstract

Employing a class of generalized connections, we describe certain differential complices $\left(\tilde \Omega^*_{\mathbb{T}}(M), \tilde{\mathbb{d}}^{\mathbb{T}}\right)$ constructed from $\wedge^* \mathbb{T} M$ and study some of their basic properties, where $\mathbb{T} M = T M \oplus T^*M$ is the generalized tangent bundle on $M$. A number of classical geometric notions are extended to $\mathbb{T} M$, such as the curvature tensor for a generalized connection. In particular, we describe an analogue to the Levi-Civita connection when $\mathbb{T} M$ is endowed with a generalized metric and a structure of exact Courant algebroid. We further describe in generalized geometry the analogues to the Chern-Weil homomorphism, a Weitzenb\"ock identity, the Ricci flow and Ricci soliton, the Hermitian-Einstein equation and the degree of a holomorphic vector bundle. Furthermore, the Ricci flows are put into the context of geometric Lax flows, which may be of independent interest.