On generalized metric structures

Let $M$ be a smooth manifold, let $TM$ be its tangent bundle and $T^{\ast }M$ 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 $M$, $E=TM\oplus T^{\ast }M$. 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 Kähler structures are also discussed. We also describe relationship between generalized metrics and weak metric structures.