Eigenvalues of Ricci Operator of Four-Dimensional Locally Homogeneous Riemannian Manifolds with Nontrivial Isotropy Subgroup

Abstract

The topology of Riemannian manifolds can be linked to the eigenvalues of curvature operators, which was demonstrated in the works of J. Milnor, V.N. Berestovsky, V.V. Slavkii, E.D. Rodionov, and Yu.G. Nikonorov. J. Milnor studied the eigenvalues of the Ricci curvature operator of left-invariant Riemannian metrics on Lie groups, and identified possible signatures of the Ricci operator for three-dimensional Lie groups. O. Kowalski and S. Nikcevic later resolved the problem of prescribed spectrum values of the Ricci operator on three-dimensional metric Lie groups and Riemannian locally homogeneous spaces. D.N. Oskorbin, E.D. Rodionov, and O.P. Khomova also obtained similar results for the one-dimensional curvature operator and the sectional curvature operator. A.G. Kremlev and Yu.G. Nikonorov investigated the fourdimensional case and studied the possible signatures of the Ricci curvature of left-invariant Riemannian metrics on Lie groups. In this study, we aim to solve the problem of prescribed eigenvalues of the Ricci operator on locally homogeneous Riemannian manifolds with a nontrivial isotropy subgroup.