Three-dimensional Riemannian manifolds with circulant structures

TL;DR

This paper studies 3D Riemannian manifolds equipped with circulant metric and endomorphism structures, exploring their curvature properties and providing explicit examples of such manifolds.

Contribution

It introduces a new class of 3D Riemannian manifolds with circulant structures and analyzes their curvature characteristics.

Findings

Derived curvature properties of the manifolds

Established compatibility conditions between metric and endomorphism

Provided explicit example of a manifold with circulant structures

Abstract

We consider a 3-dimensional Riemannian manifold M with two circulant structures -- a metric g and an endomorphism q whose third power is identity. The structure q is compatible with g such that an isometry is induced in any tangent space of M. We obtain some curvature properties of this manifold (M, g, q) and give an explicit example of such a manifold.