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

Abstract

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.

On the Sampson Laplacian

“…In the third and fourth paragraphs of the paper we consider the properties of the Sampson operator acting on one-forms and symmetric two-tensors. Theorems and corollaries of the present paper complement our results from the papers [3,31,33,39,41,42,44].…”

“…The authors of these papers and monograph have determined this operator and obtained its Weitzenböck decomposition but did not quote the source [38]. On the other hand, we were the first and only who began to study the properties of this operator in details (see [3,31,33,39,41,42,44]). To these lists, we can add two papers [24] and [25] in which there are terms "Sampson Laplacian" and "Sampson operator" but there are no new results on the geometry of the Sampson operator.…”

On a constant curvature statistical manifold

“…Remark 2.9. It should be remarked that the sufficient part of Theorem 2.8 has been proved in [16] (see [17,Theorem 2]) and [11,Theorem 3.3].…”

“…Moreover, for a trace-free statistical manifold, which is an analogue to Blaschke immersions of affine differential geometry, we will show that they are also equivalent to the conjugate symmetry of the Ricci curvature and the projective flatness of ∇ * , Theorem 2.10. It should be remarked that the sufficient part of Theorem 2.8 has been proved in [16] (see [17,Theorem 2]) and [11,Theorem 3.3].…”

Conformal Control Tools for Statistical Manifolds and for γ-Manifolds

“…In [11], 1-conformally equivalent statistical manifolds are characterized. Topological properties of some five-dimensional compact, conformally flat statistical manifolds are found in [12]. Conformal submersions with horizontal distribution and associated statistical structures were defined and characterized in [13].…”