Cyclic homogeneous Riemannian manifolds

TL;DR

This paper characterizes and classifies cyclic homogeneous Riemannian manifolds, especially in low dimensions, and provides examples of non-compact irreducible Riemannian 3-symmetric spaces with cyclic metrics.

Contribution

It offers new characterizations, classifications, and explicit examples of cyclic homogeneous Riemannian manifolds, expanding understanding in spin geometry.

Findings

Classification of simply-connected cyclic homogeneous Riemannian manifolds up to dimension four

Characterization of cyclic and traceless cyclic homogeneous Riemannian manifolds

Examples of non-compact irreducible Riemannian 3-symmetric spaces with cyclic metrics

Abstract

In spin geometry, traceless cyclic homogeneous Riemannian manifolds equipped with a homogeneous spin structure can be viewed as the simplest manifolds after Riemannian symmetric spin spaces. In this paper, we give some characterizations and properties of cyclic and traceless cyclic homogeneous Riemannian manifolds and we obtain the classification of simply-connected cyclic homogeneous Riemannian manifolds of dimension less than or equal to four. We also present a wide list of examples of non-compact irreducible Riemannian $3$-symmetric spaces admitting cyclic metrics and give the expression of these metrics.