On the Ricci operator of locally homogeneous Lorentzian 3-manifolds

Abstract

Abstract:We determine the admissible forms for the Ricci operator of three-dimensional locally homogeneous Lorentzian manifolds. MSC:53C50, 53C20, 53C30

Quasihomogeneous three-dimensional real-analytic Lorentz metrics do not exist

“…In [6] Calvaruso and Kowalski calculate Ricci operators for left-invariant Lorentz metrics on three-dimensional Lie groups, assuming they are not symmetric (see also previous curvature calculations in [5,7,19]). If the metric on U \S were symmetric, then the covariant derivative of the curvature would vanish on all of U , which would imply U locally symmetric, hence locally homogeneous; therefore, we need consider only nonsymmetric left-invariant metrics here.…”

“…On the other hand, recall that in [6] Calvaruso and Kowalski classified Ricci operators for left-invariant Lorentz metrics g on three-dimensional Lie groups. In particular, they proved (see their Theorems 3.5, 3.6 and 3.7) that a Ricci operator of a left-invariant Lorentz metric on a nonunimodular three-dimensional Lie group admits a triple eigenvalue r = 0 if and only if g is of constant sectional curvature.…”

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

“…В случае левоинвариантных лоренцевых метрик на трех-мерных группах Ли известна работа Дж. Каль-варузо, О. Ковальского [2], в которой исследуется задача о существовании группы Ли с левоинвари-антной лоренцевой метрикой и заданными значе-ниями спектра оператора Риччи.…”

Investigation of Curvature Operatorson Three-Dimensional Locally Homogeneous Lorentzian Manifolds with Application of Symbolic Computations Packages

“…В случае лоренцевых метрик на трехмерных ло-кально однородных пространствах известна рабо-та Дж. Кальварузо, О. Ковальского [2], в кото-рой исследуется задача о существовании локально однородного лоренцева пространства с заданным оператором Риччи.…”