On the Space of 3-dimensional Homogeneous Riemannian Manifolds

TL;DR

This paper investigates the existence of 3-dimensional homogeneous Riemannian manifolds with prescribed Ricci eigenvalues, addressing a fundamental question in differential geometry.

Contribution

It provides a classification and characterization of such manifolds based on their Ricci tensor eigenvalues, advancing understanding of homogeneous Riemannian geometry.

Findings

Existence conditions for manifolds with given Ricci eigenvalues

Complete classification of 3D homogeneous Riemannian manifolds with specified Ricci spectra

New insights into the structure of homogeneous spaces in relation to Ricci curvature

Abstract

We answer the following question: Let l, m, n be arbitrary real numbers. Does there exist a 3-dimensional homogeneous Riemannian manifold whose eigenvalues of the Ricci tensor are just l, m and n ?