Generalized quasi-statistical structures

TL;DR

This paper explores generalized quasi-statistical structures on manifolds, establishing their integrability conditions, inducing structures on tangent and cotangent bundles, and connecting them with various geometric metrics and Norden structures.

Contribution

It introduces the concept of generalized quasi-statistical structures, linking them with classical quasi-statistical manifolds and extending these structures to tangent and cotangent bundles.

Findings

Generalized complex and product structures are integrable iff the manifold is quasi-statistical.

Any quasi-statistical structure induces generalized quasi-statistical structures on tangent and cotangent bundles.

Prolongation of structures is possible when the connection is flat, with applications to Norden and Para-Norden 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 $TM\oplus T^*M$ 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 $TM\oplus T^*M$. 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 $T^*M$, 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 $T^*M$ and $TM$.