Riv. Mat. Univ. Parma, Vol. 13, No. 2, 2022

Adara M. Blaga [a] and Antonella Nannicini [b]

Statistical structures, α-connections and Generalized Geometry

Pages: 283-296
Received: 22 September 2021
Accepted: 24 May 2022
Mathematics Subject Classification: 53C15, 53C05, 53C38.
Keywords: Statistical structures, α-connections, generalized geometry.
Authors address:
[a]: West University of Timisoara, Department of Mathematics, Timisoara, Romania
[b]: University of Florence, Dipartimento di Matematica e Informatica “U. Dini”, Florence, Italy

This research was partially supported by the PRIN 2017 project ''Real and Complex Manifolds: Topology, Geometry and holomorphic dynamics'' (code 2017JZ2SW5)

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Abstract: The main purpose of this paper is to describe how statistical structures fit perfectly into Generalized Geometry. Firstly, we will briefly present the properties of generalized pseudo-calibrated almost complex structures induced by statistical structures. Then we will characterize the integrability of generalized almost complex structures with respect to the bracket defined by the \(\alpha\)-connection, finding conditions under which the concept of integrability is \(\alpha\)-invariant. Finally, we consider a pair of generalized dual quasi-statistical connections \((\hat{\nabla},\hat{\nabla}^*)\) on the generalized tangent bundle \(TM\oplus T^*M\) and we provide conditions for \(TM\oplus T^*M\) with the \(\alpha\)-connections \((\hat{\nabla}^{(\alpha)},\hat{\nabla}^{(-\alpha)})\) induced by \((\hat{\nabla},\hat{\nabla}^*)\) to be conjugate Ricci-symmetric.

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