Bochner’s technique for statistical structures

Abstract

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.

Curvature bounded conjugate symmetric statistical structures with complete metric

“…Moreover, the assumption about the constant curvature is replaced by the assumption that the curvature satisfies some inequalities. Since the notion of the sectional ∇-curvature is relatively new, see [6], [1], the theorems proved in this paper show that the notion is meaningful.…”

“…We shall shortly recall basic notions of statistical geometry. For details we refer to [6]. Let g be a positive definite Riemannian tensor field on a manifold M .…”

“…Sectional ∇-curvature. The notion of a sectional ∇-curvature was first introduced in [6]. Namely, the tensor field…”

“…In general, this condition is not satisfied by the curvature tensor R. In the case where a given statistical structure is conjugate symmetric the curvature tensor R satisfies this condition. In [6] we defined the sectional ∇-curvature by (17) k(π) = g(R(e 1 , e 2 )e 2 , e 1 ) for a vector plane π ∈ T x M , x ∈ M and e 1 , e 2 any orthonormal basis of π. It is a well-defined notion.…”

Generalized Wintgen inequality for statistical submanifolds in statistical manifolds of constant curvature

“…In this paper, using the sectional curvature defined in [19], we derive a generalized Wintgen inequality for statistical submanifolds in statistical manifolds of constant curvature.…”