Neutral Ricci spectra of homogeneous Riemannian four-manifolds

Abstract

It is shown that every quadruple of real numbers with two positive and two negative entries is the Ricci spectrum of some left-invariant Riemannian metric on a Lie group of dimension four. Consequently, for eight out of the fifteen possible Ricci signatures of four-dimensional homogeneous Riemannian manifolds, one can determine precisely which spectra in each given signature actually occur. We also describe the Ricci spectra realized by left-invariant metrics on a certain class of 2-step solvable Lie groups of all dimensions $n>2$.