On Ricci eigenvalues of locally homogeneous Riemannian 3-manifolds

On the Ricci operator of locally homogeneous Lorentzian 3-manifolds

“…Another remarkable difference between the Riemannian and Lorentzian case is given by the fact that in the Riemannian one, an orthonormal basis of g, diagonalizing L, also diagonalizes the Ricci operator [11,13]. However, as we shall see, in the Lorentzian case, different Segre types for the Ricci operator occur for the same Segre type of L, making the classification harder but also more interesting.…”

“…for all ∈ g. This case was already investigated by Nomizu [15], who proved that any Lorentzian metric on a Lie group G, whose Lie algebra satisfies (11), has constant sectional curvature, and this constant can be any real number (see Theorem 1 of [15]). In particular, G is symmetric.…”

“…Let G be a three-dimensional Lie group, equipped with a left-invariant pseudoRiemannian metric and having a non-unimodular Lie algebra g which does not satisfy (11). We denote by the (two-dimensional) unimodular kernel of g, defined by = Ker{tr ad : g → R}…”

“…Because of this result, the problem essentially reduces to determining precisely which Segre types occur for all three-dimensional Lorentzian Lie groups and all Lorentzian symmetric 3-spaces. The corresponding study in the Riemannian framework was made by the second author and S. Nikčević in [11], where the range of all real triplets occurring as Ricci eigenvalues of a locally homogeneous Riemannian 3-space was determined. The paper is organized in the following way.…”

On the Sectional Curvature Operator of Three-Dimensional Lie Groups with Left-Invariant Lorentzian Metrics

“…Задачи о восстановлении (псевдо)риманова многообразия по предписанному спектру оператора кривизны являются актуальным направлением в исследова-МАТЕМАТИКА нии операторов кривизны. Римановы локально-однородные пространства с предписанными зна-чениями спектра оператора Риччи были опреде-лены О. Ковальским и С. Никшевич в [1]. В случае левоинвариантных лоренцевых метрик на трех-мерных группах Ли известна работа Дж.…”