On the geometry of four dimensional Riemannian manifold with a circulant metric and a circulant affinor structure

TL;DR

This paper explores the geometric properties of a four-dimensional Riemannian manifold with a circulant metric and affinor structure, analyzing curvature relations and the existence of special bases under certain conditions.

Contribution

It introduces the concept of a q-base in the tangent space and establishes curvature equalities when the affinor structure is covariantly constant.

Findings

Existence of a q-base in the tangent space.

Curvature equalities when abla q=0.

Relations among sectional curvatures.

Abstract

We consider a four dimensional Riemannian manifold M with a metric g and an affinor structure q. We note the local coordinates of g and q are circulant matrices. Their first orders are (A, B, C, B), A, B, C \in FM and (0, 1, 0, 0), respectively. Let \nabla be the connection of g. Further, let mu_{1}, mu_{2},mu_{3}, mu_{4}, mu_{5}, mu_{6} be the sectional curvatures of 2-sections {x, qx}, {x, q^{2}x}, {q^{3}x, x}, {qx, q^{2}x}, {qx, q^{3}x}, {q^{2}x, q^{3}x} for arbitrary vector x in T_{p}M$, p is in M . Then we have that q^{4}=E; g(qx, qy)=g(x,y), x, y are in chiM. The main results of the present paper are 1) There exist a q-base in T_{p}M, p is in M. 2) if \nabla q=0, then \mu_{1}= \mu_{3}=\mu_{4}=\mu_{6}, \mu_{2}= \mu_{5}=0.