On affine connections in a Riemannian manifold with a circulant metric and two circulant affinor structures

TL;DR

This paper explores the geometric properties of a specific class of 3-dimensional Riemannian manifolds with circulant metric and affinor structures, deriving relations between their curvature tensors and curvatures.

Contribution

It introduces a new class of manifolds with circulant tensors and establishes relations between their curvature tensors and sectional curvatures.

Findings

Derived relation between curvature tensors R and ar{R}

Identified conditions when ar{R} vanishes and R simplifies

Connected sectional curvature of specific 2-sections to scalar curvature

Abstract

In the present paper it is considered a class V of 3-dimensional Riemannian manifolds M with a metric g and two affinor tensors q and S. It is defined another metric \bar{g} in M. The local coordinates of all these tensors are circulant matrices. It is found: 1)\ a relation between curvature tensors R and \bar{R} of g and \bar{g}, respectively; 2)\ an identity of the curvature tensor R of g in the case when the curvature tensor \bar{R} vanishes; 3)\ a relation between the sectional curvature of a 2-section of the type \{x, qx\} and the scalar curvature of M.