Curvature properties of $4$-dimensional Riemannian manifolds with a circulant structure

TL;DR

This paper investigates the curvature properties of 4-dimensional Riemannian manifolds equipped with a circulant structure, providing theoretical results, an explicit example on a Lie group, and analysis of sectional curvatures.

Contribution

It introduces and analyzes the curvature properties of 4D Riemannian manifolds with a circulant structure, including explicit construction and geometric characteristics.

Findings

Derived conditions for sectional curvatures of 2-planes.

Constructed an explicit example on a Lie group.

Identified geometric characteristics of the example.

Abstract

We consider a $4$-dimensional Riemannian manifold $M$ equip\-ped with a circulant structure $q$, which is an isometry with respect to the metric $g$ and $q^{4}=\id$, $q^{2}\neq \pm \id$. For such a manifold $(M, g, q)$ we obtain some assertions for the sectional curvatures of $2$-planes. We construct an example of such a manifold on a Lie group and we find some of its geometric characteristics.