Einstein metrics from symmetry and bundle constructions (Q2771298)

In this article the author surveys only the papers on Einstein metrics, published in the last decade, whose contributions the author knows how to link into a coherent whole. Recall that the Einstein equation \(\text{Ric}(g)=\Lambda g\) is a nonlinear second order system of partial differential equations which is invariant under the action of the diffeomorphism group of the manifold. The symmetry method means constructing Einstein metrics having a finite dimensional Lie group of isometries. If this Lie group acts transitively or acts with hypersurface principal orbits, the Einstein equation becomes respectively a system of algebraic or ordinary differential equations. The method of bundle structures consists in constructing Einstein metrics on the total space of bundles which are put together from special families of metrics on the fibres and the base using suitable connections. The author discusses mainly the construction of Einstein metrics whose holonomy group is generic, i.e., the restricted holonomy is \(SO(n)\), where \(n\) is the dimension of the manifold. NEWLINENEWLINENEWLINEThe paper consists of four sections. In Section one the author reminds us the Kaluza-Klein construction on principal \(G\)-bundles. Let \(\pi :P\rightarrow M\) be a smooth principal \(G\)-bundle with \(G\) a compact Lie group and \(\Phi \) a connection on \(P\) with curvature form \(\Omega \). If \(\langle\;,\;\rangle \) is a left-invariant metric on \(G\) and \(g^{*}\) a metric on \(M\), then we can construct a metric \(g\) on \(P\) by the formula \(g(X,Y)=g^{*}(\pi _{*}X,\pi _{*}Y)+\langle \Phi (X),\Phi (Y)\rangle .\) The connection \(\Phi \) is said to be Yang-Mills if \(\Omega \) is coclosed as an \(ad(g)\)-valued 2-form on \(M\). The system of partial differential equations equivalent with the Einstein condition for \(g\) is derived. Recall that Einstein condition implies that \(g^{*}\) has constant scalar curvature and the pointwise norm of \(\Omega \) must be constant. The case of an abelian group \(G\) is first analyzed.NEWLINENEWLINENEWLINEThe Einstein metrics on principal torus bundles with given characteristic classes are determined and the results of J. Wang concerning the Einstein metrics on principal circle bundles are presented. Some examples of Einstein metrics that can be constructed by solving a system of algebraic equations in certain scaling parameters are given. Then, some examples with non-abelian group \(G\) and some Kaluza-Klein constructions on fibre bundles associated with principal bundles are given, too. Recall that a connected \(G\)-manifold is said to be of cohomogeneity 1 if the principal orbits are hypersurfaces. The orbit space of a cohomogeneity 1 manifold is either an interval \(I\) or a circle. The second section discusses Einstein metrics of cohomogeneity 1, i.e. Einstein metrics on \(G\)-manifolds of cohomogenity 1. In the former situation, the Einstein condition reduces to a system of nonlinear ordinary differential equations on \(I\) together with boundary conditions to insure that we have a smooth metric. The initial value problem for this system is discussed.NEWLINENEWLINENEWLINEThe hyper-Kähler and quaternionic Kähler metrics of cohomogeneity 1 are analyzed and some classifying results are given. For example, Dancer and Swann proved that a complete quaternionic-Kähler manifold with positive scalar curvature which has a semisimple compact group of isometries with cohomogeneity 1 must be quaternionic symmetric. The examples with special holonomy can be treated by symmetry method, while the examples with generic holonomy can be treated by bundle methods. Böhm proved that there exist infinitely many non-isometric Einstein metrics of cohomogenity 1 on \(M=S^{p+1}\times Q^{q}\) where 5\(\leq p+q+1\leq 9,\) \(p>1,\) \(q>1\) and \(Q\) is a non-flat compact isotropy irreducible homogeneous space. The non-existence criterion of Böhm for cohomogeneity 1 Einstein metrics on closed manifolds is given. The Kaluza-Klein ansatz is the construction of Einstein metric \(g\) on \(P\) by the formula presented before. NEWLINENEWLINENEWLINEIn section 3 a modification of the Kaluza-Klein ansatz on fibre bundles is considered. A special case that gives rise to large families of Einstein Hermitian metrics and also unifies and generalizes many known examples is discussed. In particular, one obtains some Einstein metrics on certain Fano manifolds when Kähler-Einstein metrics are obstructed. The modified Kaluza-Klein ansatz can be used to construct Einstein metrics with special holonomy (such as of type \(G_{2}\) or Spin(7)). NEWLINENEWLINENEWLINEThe last section deals with homogeneous Einstein metrics. Homogeneous Einstein metrics on a compact homogeneous space \(G/K\) are the critical points of the scalar curvature function on the space of \(G\)-invariant metrics with a fixed value for the volume. Both homogeneous Einstein metrics with positive and negative sectional curvature are analyzed.NEWLINENEWLINEFor the entire collection see [Zbl 0961.00021].