Differential calculus for generalized geometry and geometric Lax flows

Shengda Hu

Vol. 331 (2024), No. 1, 23–76

DOI: 10.2140/pjm.2024.331.23

Abstract

It is of interest to extend classical geometric notions to generalized geometry. Various approaches have been proposed in the recent literature. Employing a class of generalized connections, we describe certain differential complices $({\tilde{\mathrm{\Omega }}}_{\mathbb{𝕋}}^{\ast }(M),{\tilde{}}^{\mathbb{𝕋}})$ constructed from ${\wedge <!--FUNCTION\; APPLICATION-->}^{\ast }\mathbb{𝕋}M$ and study some of their basic properties, where $\mathbb{𝕋}M=TM\oplus T^{\ast }M$ is the generalized tangent bundle on $M$. To illustrate how various constructions fit together from this point of view, we describe within the proposed framework the analogues to the Levi-Civita connection when $\mathbb{𝕋}M$ is endowed with a generalized metric and a structure of exact Courant algebroid, the Chern–Weil homomorphism, a Weitzenböck identity, the Ricci flow as a Lax flow and Ricci soliton, the Hermitian–Einstein equation and the degree of a holomorphic vector bundle.

Keywords

generalized connections, de Rham cohomology, generalized curvature, geometric Lax flows, generalized Kähler geometry, generalized holomorphic bundles

Mathematical Subject Classification

Primary: 53B15, 53B20, 53D18, 53E99

Milestones

Received: 6 September 2022

Revised: 9 May 2024

Accepted: 12 July 2024

Published: 2 October 2024