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 the prolongations of the quasi-statistical structures on the manifolds to their cotangent and tangent bundles via Generalized Geometry. Moreover, Norden and Para-Norden structures are defined on $T^*M$ and $TM$.
Citation
Adara M. Blaga. Antonella Nannicini. "Generalized quasi-statistical structures." Bull. Belg. Math. Soc. Simon Stevin 27 (5) 731 - 754, december 2020. https://doi.org/10.36045/j.bbms.191023
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