Dual symmetric statistical manifolds

Stepanova, Elena S.

doi:10.1007/s10958-007-0488-4

Cited by 8 publications

(5 citation statements)

References 2 publications

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“…On one side, our paper shows new geometrical interpretation to the theory of abstract statistical manifolds, on the other side, we study new geometrical and topological properties of five-dimensional manifolds with SO(3)-structures. This paper is a continuation of studies of abstract statistical manifolds, which the second author began in her papers [49,50].…”

Section: Introductionmentioning

confidence: 94%

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A five-dimensional Riemannian manifold with an irreducible SO(3)-structure as a model of abstract statistical manifold

Mikeš

Stepanova

2013

Ann Glob Anal Geom

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In the present paper, we consider a five-dimensional Riemannian manifold with an irreducible SO(3)-structure as an example of an abstract statistical manifold. We prove that if a five-dimensional Riemannian manifold with an irreducible SO(3)-structure is a statistical manifold of constant curvature, then the metric of the Riemannian manifold is an Einstein metric. In addition, we show that a five-dimensional Euclidean sphere with an irreducible SO(3)-structure cannot be a conjugate symmetric statistical manifold. Finally, we show some results for a five-dimensional Riemannian manifold with a nearly integrable SO(3)-structure. For example, we prove that the structure tensor of a nearly integrable SO(3)-structure on a five-dimensional Riemannian manifold is a harmonic symmetric tensor and it defines the first integral of third order of the equations of geodesics. Moreover, we consider some topological properties of five-dimensional compact and conformally flat Riemannian manifolds with irreducible SO(3)-structure.

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Section: Introductionmentioning

confidence: 94%

“…165-215] gave a modern differential geometric treatment of statistical problems by introducing the notion of an abstract statistical manifold. This generalization has been considered in many papers by other geometers ( [6,24,29,30,49,50] and etc.). In this section, we give an overview of the geometry of such manifold.…”

Section: An Abstract Statistical Manifoldmentioning

confidence: 99%

A five-dimensional Riemannian manifold with an irreducible SO(3)-structure as a model of abstract statistical manifold

Mikeš

Stepanova

2013

Ann Glob Anal Geom

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“…where U is a 1-form and ϕ is a tensor of type (1,1). Afterwards, several types of a quarter-symmetric metric connection were studied ( [8,15,21,25,28]). In [11,20,22,26,31,32], the geometric and physic properties of conformal and projective the semi-symmetric metric recurrent connections were studied.…”

Section: Introductionmentioning

confidence: 99%

Geometric characteristics of a manifold with a symmetric-type quarter-symmetric projective conformal non-metric connection

Zhao

Kwak

et al. 2021

Filomat

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“…In [7] a curvature copy problem of a nonmetric connection was considered. In [16] the Amari-Chentsov connection as a geometrical structure of a statistical manifold was regarded and in [18] a conjugate symmetry condition of the statistical manifold was regarded and in [18] a conjugate symmetry condition of the statistical manifold discovered. In [1,2,4,5] and [17] the properties of a semi-symmetric non-metric connection were investigated.…”

Section: Introductionmentioning

confidence: 99%