On generalized metric structures

TL;DR

This paper explores integrability conditions for various generalized geometric structures on the tangent bundle of a manifold, providing criteria based on forms and connections, with implications for pseudo-Riemannian and Kähler geometries.

Contribution

It introduces new integrability criteria for generalized metrics and structures, linking properties of forms and connections, and extends results to pseudo-Riemannian contexts.

Findings

Provided sufficient conditions for integrability of generalized structures.

Connected generalized metrics with weak metric structures.

Extended integrability criteria to pseudo-Riemannian settings.

Abstract

Let $M$ be a smooth manifold, let $TM$ be its tangent bundle and $T^{*}M$ its cotangent bundle. This paper investigates integrability conditions for generalized metrics, generalized almost para-complex structures, and generalized Hermitian structures on the generalized tangent bundle of $M$, $E=TM \oplus T^{*}M$. In particular, two notions of integrability are considered: integrability with respect to the Courant bracket and integrability with respect to the bracket induced by an affine connection. We give sufficient criteria that guarantee the integrability for the aforementioned generalized structures, formulated in terms of properties of the associated $2$-form and connection. Extensions to the pseudo-Riemannian setting and consequences for generalized Hermitian and K\"ahler structures are also discussed. We also describe relationship between generalized metrics and weak metric structures.