Generalized quasi-statistical structures

2018/09/13 by Blaga, Adara M., Nannicini, Antonella
#53B05 #53C15 #53D05 #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1809.04784

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⊕ T^*M of M and we show that they are ∇-integrable, for ∇ an affine connection on M, if and only if (M,h,∇) 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⊕ 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 ∇ 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.

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