$\alpha$-connections in generalized geometry

Abstract

We consider a family of $\alpha$-connections defined by a pair of generalized dual quasi-statistical connections $(\hat{\nabla },{\hat{\nabla }}^{\ast })$ on the generalized tangent bundle $(TM\oplus T^{\ast }M,\check{h})$ and determine their curvature, Ricci curvature and scalar curvature. Moreover, we provide the necessary and sufficient condition for ${\hat{\nabla }}^{\ast }$ to be an equiaffine connection and we prove that if $h$ is symmetric and $\nabla h=0$, then $(TM\oplus T^{\ast }M,\check{h},{\hat{\nabla }}^{(\alpha )},{\hat{\nabla }}^{(-\alpha )})$ is a conjugate Ricci-symmetric manifold. Also, we characterize the integrability of a generalized almost product, of a generalized almost complex and of a generalized metallic structure w.r.t. the bracket defined by the $\alpha$-connection. Finally we study $\alpha$-connections defined by the twin metric of a pseudo-Riemannian manifold, $(M,g)$, with a non-degenerate $g$-symmetric $(1,1)$-tensor field $J$ such that $d^{\nabla }J=0$, where $\nabla$ is the Levi-Civita connection of $g$.

29 pages

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