Flat Generalized Connections on Courant Algebroids
Gil R. Cavalcanti, Jaime Pedregal, Roberto Rubio
TL;DR
This paper investigates flat generalized Levi-Civita connections on transitive Courant algebroids, focusing on the Bismut family of metric generalized connections that interpolate between canonical Levi-Civita and Gualtieri–Bismut connections. It establishes that, for a flat generalized Bismut connection with nonzero closed 3-form H on a 1-connected, complete, irreducible Riemannian manifold, the base is a compact simple Lie group with a bi-invariant metric and a Cartan 3-form. It then explicitly describes the space of flat left-invariant generalized Levi-Civita connections on a Lie group, showing this space is strictly smaller than the full space of Levi-Civita generalized connections and that non-flat examples exist. Together, these results illuminate the generalized holonomy and rigidity phenomena in Bismut geometry and provide a concrete classification in the Lie-group setting.
Abstract
We consider a family of metric generalized connections on transitive Courant algebroids, which includes the canonical Levi-Civita connection, and study the flatness condition. We find that the building blocks for such flat transitive Courant algebroids are compact simple Lie groups. Further, we give a description of left-invariant flat Levi-Civita generalized connections on such Lie groups, which, in particular, shows the existence of non-flat ones.