Einstein Metrics And Einstein-Randers Metrics On Some Homogeneous Spaces

In this thesis,we mainly study Einstein metrics and Einstein-Randers metrics on some homogeneous spaces.Firstly,we study invariant Einstein-Randers metrics on the Stiefel manifold V2pRn,which is diffeomorphic to the homogeneous space SO(n)/SO(n2p).Based on some results of Einstein metrics,we construct invariant Einstein-Randers metrics on the Stiefel manifold SO(n)/SO(n-2p).Secondly,we consider invariant Einstein and Einstein-Randers metrics on certain homogeneous spaces of exceptional type,which are arising from flag manifolds with two isotropy summands.For these homogeneous spaces,we view the homogeneous spaces as total spaces over some flag manifolds.Then we consider a special G invariant metric on the flag manifolds.Based on some results,we can deduce the homogeneous Einstein equations for each case.By the aids of computer,we can solve the corresponding system of equations,then we can construct new Einstein metrics on these homogeneous spaces.Furthermore,we consider Einstein-Randers metrics on the homogeneous manifolds,based on the discussion in Riemannian case,we construct new examples of Einstein-Randers metrics.Finally,we study invariant Einstein metrics and Einstein-Randers metrics on homogeneous spaces of exceptional type arising from generalized Wallach spaces.We consider a special G invariant metric,with the structure coefficients,we obtain the system of homogeneous Einstein equations with respect to these metrics,then we get new invariant Einstein metrics.Furthermore,we constructed new Einstein-Randers metrics for each type.