Einstein Metrics on Bundles

Abstract

<p>Given a principal circle bundle with a nontrivial connection over a compact Kähler manifold, there is a Riemannian metric on the bundle by horizontal lift of the metric from the base. It is shown that, if the curvature form of the connection is of type (1,1) and the bundle is Einstein, then the base is a finite product of Kähler-Einstein manifolds with positive first Chern classes and the Euler class of the bundle must be a linear combination of the first Chern classes of the manifolds in the base. This is the uniqueness of a construction of Einstein metrics given by M.Y. Wang and W. Ziller. It is pointed out that the uniqueness theorem also holds for a class of principal torus bundles and some uniqueness for principal bundles with non-abelian structure groups are given as well. The existence of Einstein metrics is given on some S²-bundles. Given finitely many Kähler-Einstein manifolds with positive first Chern classes, we have principal circle bundles over their product whose Euler classes are linear combinations of their first Chern classes. To every such circle bundle an S²-bundle can be associated. I show that there are at least two families of Einstein metrics on these S²-bundles. For those with positive first Chern classes with respect to a natural complex structure on the total space, I present non-Kählerian Einstein metrics when the Futaki obstruction to the existence of Kähler-Einstein metrics does not vanish. An antipodal identification of every fibre of an S²-bundle will yield an RP² bundle and the other family of Einstein metrics are lifts of those constructed on the associated RP² bundles. Finally, the author suggests new constructions of Einstein metrics on certain principal S¹-bundles which allow the total spaces to be even dimensional.</p>