On 3-dimensional almost Einstein manifolds with circulant structures

TL;DR

This paper investigates 3-dimensional almost Einstein manifolds with circulant structures, classifying them into three types based on curvature properties and parallelism, and provides geometric characteristics and examples.

Contribution

It introduces a classification of 3D almost Einstein manifolds with circulant structures and analyzes their geometric properties and examples.

Findings

Three classes of manifolds identified based on curvature and parallelism.

Explicit geometric characteristics derived for each class.

Examples illustrating the theoretical classifications provided.

Abstract

A 3-dimensional Riemannian manifold equipped with a tensor structure of type $(1,1)$, whose third power is the identity, is considered. This structure and the metric have circulant matrices with respect to some basis, i.e., these structures are circulant. An associated manifold, whose metric is expressed by both structures, is studied. Three classes of such manifolds are considered. Two of them are determined by special properties of the curvature tensor of the manifold. The third class is composed by manifolds whose structure is parallel with respect to the Levi-Civita connection of the metric. Some geometric characteristics of these manifolds are obtained. Examples of such manifolds are given.