Four-dimensional Riemannian manifolds with two circulant structures

TL;DR

This paper studies four-dimensional Riemannian manifolds with a circulant structure q, exploring their curvature properties, conditions for q to be parallel, and the geometric implications of these structures.

Contribution

It introduces a subclass of manifolds with circulant structures, providing conditions for q to be parallel and analyzing sectional curvatures within this framework.

Findings

Derived necessary and sufficient conditions for q to be parallel.

Established relations between the curvature tensor and circulant structures.

Analyzed sectional curvatures of specific two-planes.

Abstract

We consider a class (M, g, q) of four-dimensional Riemannian manifolds M, where besides the metric g there is an additional structure q, whose fourth power is the unit matrix. We use the existence of a local coordinate system such that there the coordinates of g and q are circulant matrices. In this system q has constant coordinates and q is an isometry with respect to g. By the special identity for the curvature tensor R generated by the Riemannian connection of g we define a subclass of (M, g, q). For any (M, g, q) in this subclass we get some assertions for the sectional curvatures of two-planes. We get the necessary and sufficient condition for g such that q is parallel with respect to the Riemannian connection of g.