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Bochner’s technique for statistical structures

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Abstract

The main aim of this paper is to extend Bochner’s technique to statistical structures. Other topics related to this technique are also introduced to the theory of statistical structures. It deals, in particular, with Hodge’s theory, Bochner–Weitzenböck and Simon’s type formulas. Moreover, a few global and local theorems on the geometry of statistical structures are proved, for instance, theorems saying that under some topological and geometrical conditions a statistical structure must be trivial. We also introduce a new concept of sectional curvature depending on statistical connections. On the base of this notion we study the curvature operator and prove some analogues of well-known theorems from Riemannian geometry.

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Acknowledgments

The research supported by the NCN Grant UMO-2013/11/B/ST1/02889 and a grant of the TU Berlin.

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Authors and Affiliations

  1. Faculty of Mathematics and Computer Science, Jagiellonian University, ul. Łojasiewicza 6, 30-348, Cracow, Poland

    Barbara Opozda

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  1. Barbara Opozda
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Correspondence to Barbara Opozda.

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Opozda, B. Bochner’s technique for statistical structures. Ann Glob Anal Geom 48, 357–395 (2015). https://doi.org/10.1007/s10455-015-9475-z

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  • DOI: https://doi.org/10.1007/s10455-015-9475-z

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