Conformal Vector Fields And Einstein Metrics On Lie Groups And Related Topics

This thesis contains three topics:invariant conformal vector fields,left-invariant pseudo-Riemannian Einstein metrics on Lie groups and the moment map for the variety of the 3-Lie algebras.For the first topic,we study two classes invariant conformal vector fields on pseudo-Riemannian Lie groups(G,<·,·>)of signature(p,q),i.e.,(i)non-Killing left-invariant vector fields;(ii)conformal vector fields induced by derivations.For non-Killing left-invariant vector fields:Firstly,we prove that the pseudo-Riemannian Lie group(G,<·,·>)is essential if(G,<·,·>)admits a non-Killing left-invariant vector field.Sec-ondly,we define an integer d=dim[g,g]dim g+min{p,q },and show that if d≤1,then g is solvable,where g denotes the Lie algebra of G.In particular,it implies that a Lorentzian(min {p,q }=1)or trans-Lorentzian(min {p,q }=2)Lie group with such a vector field is solvable.Moreover,we construct non-solvable pseudo-Riemannian Lie groups that admit a non-Killing left-invariant vector field for any min {p,q }3.Third-ly,we give a complete classification of the Lorentzian case.It is proved that[g,g]is a direct sum of a generalized Heisenberg Lie algebra and an Abelian Lie algebra.Further-more,we obtain a simple criterion for such Lorentzian Lie groups with dim G 4 to be conformally flat,and examples of conformally flat and non-conformally flat Lorentzian Lie groups are both constructed.As a byproduct,we prove that Lorentzian Lie groups with non-Killing left-invariant conformal vector fields are Ricci solitons,which can be shrinking,steady and expanding.For conformal vector fields induced by derivations:Firstly,we point out that pseudo-Riemannian Lie group(G,<·,·>)with conformal vec-tor fields induced by derivations is a natural generalization of non-Killing left-invariant conformal vector fields.Moreover,we prove that if the derivation is a non-invertible,then(G,<·,·>)is essential.Secondly,we show that a Riemannian(min {p,q }=0)or Lorentzian Lie group with such a conformal vector field is solvable.And we also construct non-solvable unimodular pseudo-Riemannian Lie groups with such vector fields for any min(p,q)2.Finally,we give the classification for the Riemannian and Lorentzian cases,respectively.For the second topic,we study pseudo-Riemannian Einstein metrics on Einstein solvmanifolds.A Riemannian Einstein manifold is called an Einstein solvmanifold if there exists a transitive solvable group of isometries.We prove that every Einstein solvmanifold admits at least one non-trivial pseudo-Riemannian Einstein metric.Finally,for the moment map m:PVn!iu(n)for the action of GL(n)on (?),we study the critical points of Fn=(?).We first prove that[∈]2PVn is a critical point if and only if m([∈])=c∈I+D∈for some c∈2R and D∈2,and that there exists a constant c>0 such that the eigenvalues of c D∈are integers prime to each other and some eigenvalues can be strictly negative.Then we give a description of the maxima and minima of Fn:Ln!R,where Ln is the projectivization of all n-dimensional 3-Lie algebras.Furthermore,the structure of the critical points of Fnis discussed.Finally,as an application,we show that every three-dimensional 3-Lie algebra is isomorphic to a critical point of F3;and there exist a curve of nonisomorphic four-dimensional 3-Lie algebras which are not isomorphic to any critical point of F4.