Generalized quasi-statistical structures

Abstract

Given a non-degenerate $(0,2)$-tensor field $h$ on a smooth manifold $M$, we consider a natural generalized complex and a generalized product structure on the generalized tangent bundle $TM\oplus T^{\ast }M$ of $M$ and we show that they are $\nabla$-integrable, for $\nabla$ an affine connection on $M$, if and only if $(M,h,\nabla )$ is a quasi-statistical manifold. We introduce the notion of generalized quasi-statistical structure and we prove that any quasi-statistical structure on $M$ induces generalized quasi-statistical structures on $TM\oplus T^{\ast }M$. In this context, dual connections are considered and some of their properties are established. The results are described in terms of Patterson-Walker and Sasaki metrics on $T^{\ast }M$, horizontal lift and Sasaki metrics on $TM$ and, when the connection $\nabla$ is flat, we define prolongation of quasi-statistical structures on manifolds to their cotangent and tangent bundles via generalized geometry. Moreover, Norden and Para-Norden structures are defined on $T^{\ast }M$ and $TM$.

28 pages