Bochner统计结构技术
Bochner's technique for statistical structures
摘要：
本文的主要目的是扩大Bochner的技术统计结构。与此技术相关的其他主题也介绍了统计结构理论。它涉及，特别是与霍奇的理论，Bochner-Weitzenbock和西蒙的类型公式。此外，还证明了一些关于统计结构几何的整体和局部定理，例如，在一定的拓扑和几何条件下，统计结构必定是平凡的。我们还引入了一个新的概念，截面曲率依赖于统计连接。在此基础上，我们研究了曲率算子，并证明了一些类似的著名定理从黎曼几何。
英文摘要：
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-Weitzenbock 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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