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It is an important problem in differential geometry to find non-naturally reductive homogeneous Einstein metrics on homogeneous manifolds. In this paper, we consider this problem for some coset spaces of compact simple Lie groups. A new method to construct invariant non-naturally reductive Einstein metrics on normal homogeneous Einstein manifolds is presented. In particular, we show that on the standard homogeneous Einstein manifolds, except for some special cases, there exist plenty of such metrics. A further interesting result of this paper is that on some compact semisimple Lie groups, there exist a large number of left invariant non-naturally reductive Einstein metrics which are not product metrics.

Non-naturally reductive Einstein metrics on normal homogeneous Einstein manifolds