Left-Invariant Pseudo-Einstein Metrics On Noncompact Semisimple Lie Groups

The notion of Einstein manifold historically arose in the context of Einstein’s gen-eral theroy of relativity. Einstein manifolds are so important in mathematical physicsandpuremathematicsthatpeoplehavealwaysmaintainedastronginterestinthem. Ein-stein manifolds called by physicists usually refer to a generalized concept, i.e., pseudo-Einstein manifolds called by mathematicians (Lorentzian Einstein manifolds are a spe-cialkindofthem). Apseudo-Riemannianmanifold(M,g)issaidtobeEinstein(pseudo-Einstein respectively) if there exists a real constant λ satisfying the Einstein equation,Ric_g=λ g. Recent advances in mathematical physics make people more and more in-terested in general pseudo-Einstein manifolds.It is very difficult to obtain general results on Einstein manifolds. A large numberof known examples of Einstein manifolds come from homogeneous Riemannian mani-folds. Homogeneous Einstein manifolds can be roughly divided into three classes, i.e.,flat ones, compact ones and noncompact ones. On noncompact Homogeneous Einsteinmanifolds, Alekseevskii made a conjecture, i.e., if G/K is a noncompact homogeneousEinstein manifold, then K is a maximal compact subgroup of G. According to thisconjecture, noncompact homogeneous Einstein manifolds might be of only one type,i.e, solvmanifolds. In particular, noncompact semisimple Lie groups might admit noleft-invariant Einstein metrics.In this paper, we focus on left-invariant pseudo-Einstein metrics on noncompactsemisimple Lie groups. In contrast to the Einstein case, many left-invariant pseudo-Einstein metrics are obtained by us on noncompact semisimple Lie groups. The maintechnique we use is the correspondence between the compact Lie algebras and the non-compact real semisimple Lie algebras determined by the involutions, about which wesketch the classification theory due to Zhida Yan. To begin with, we deduce a formulaof Ricci curvatures associated with left-invariant pseudo-Riemannian metrics on Lie groups, i.e.,and give some examples of pseudo-Einstein metrics on low-dimensional Lie groups.Some examples of these show that the situation becomes completely different whenthe scope of our study is extended from Riemannian metrics to pseudo-Riemannianmetrics. We also discuss the pseudo-Einstein metrics on the low-dimensional compactLie groups. Before the construction of left-invariant pseudo-Einstein metrics on non-compact semisimple Lie groups, we prove our main theorem of the correspondence ofpseudo-Einstein metrics on the couple of Lie groups associated with a fixed involution.Since many left-invariant Einstein metrics on compact Lie groups of type A, B and Dhave been got by other persons, we construct left-invariant pseudo-Einstein metrics onnoncompact semisimple Lie groups of these types from them. Furthermore, we directlygive some additional left-invariant pseudo-Einstein metrics on compact and noncom-pact Lie groups as a supplement. At last, we discuss the possibilities and difficulties toclassify left-invariant pseudo-Einstein metrics on noncompact Lie groups especially onlow-dimensional ones.