Bochner’s technique for statistical structures
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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.
KeywordsAffine connection Statistical structure Curvature tensors Laplacian Hedge’s theory Bochner’s technique
Mathematics Subject ClassificationPrimary: 53B05 53C05 53A15 53B20
The research supported by the NCN Grant UMO-2013/11/B/ST1/02889 and a grant of the TU Berlin.