On Ricci eigenvalues of locally homogeneous Riemann 3-manifolds
A 3-dimensional Riemannian manifold \((M^3,g)\) is curvature homogeneous if and only if the principal Ricci curvatures \(\rho_i\), \(i=1,2,3\), are constant. Locally homogeneous \((M^3,g)\) are trivially curvature homogeneous. While there always exist curvature homogeneous \((M^3,g)\) with an arbitrary given triplet \((\rho_1,\rho_2,\rho_3)\), \(\rho_i\in \mathbb{R}\), it follows from the well-known Milnor results that this is not the case for locally homogeneous \((M^3,g)\). The main purpose of the paper is to determine the exact ranges of all triplets \((\rho_1,\rho_2,\rho_3) \in \mathbb{R}^3\) which can occur as Ricci eigenvalues of locally homogeneous \((M^3,g)\).
locally homogeneous Riemann three-manifolds
scientific article; zbMATH DE number 929908