The Einstein Metrics And Einstein-Randers Metrics On Some Homogeneous Manifolds

A compact simple Lie groups G whose structure and classification is clear can be viewed as principal bundles over flag manifolds M = G/K.If G have three isotropy summands and second Betti number b2(M)=1,then K=K0×K1,where K0 is one dimensional and K1 is semisimple.G/K1 can be regarded as principal S1 bundles on G/K.By a.theorem of Kobayashi in his paper,we know that every G/K1 admits an invariant Einstein metric.On the decomposition,we have that each ho-mogeneous manifold G/K1 admits three Ad(K)-invariant Einstein metrics,admits at least three families of G-invariant non-Riemannian Einstein-Randers metrics.In chapter 4,We calculate the Einstein metrics on two special homogeneous man-ifolds separately.E6/SU(5)admits at least four invariant Einstein metrics which are not isometric,admits at least four families of E6-invariant non-Riemannian E-instein-Randers metrics,E6/SU(2)admits at least two invariant Einstein metrics which are not isometric,admits at least two families of E6-invariant non-Riemannian Einstein-Randers metrics.This paper is organized into four parts.In chapter 1,we introduce the research background and progress of Einstein metrics and Einstein-Randers metrics.In chapter 2,we present some fundamental knowledge of Einstein metrics and Einstein-Randers metrics that we need in the paper.In chapter 3,we compute the Ad(K)-invariant Einstein metrics and Einstein-Randers metrics on G/K1.In chapter 4,we compute Einstein metrics and Einstein-Randers metrics on some special homogeneous spaces E6/SU(5)and E6/SU(2).