Non Naturally Reductive Einstein Metrics on the Orthogonal Group via Real Flag Manifolds

Abstract

We obtain new invariant Einstein metrics on the compact Lie groups $${{\,\textrm{SO}\,}}(n)$$ SO ( n ) which are not naturally reductive. This is achieved by using the real flag manifolds $${{\,\textrm{SO}\,}}(k_1+\cdots +k_p)/{{\,\textrm{SO}\,}}(k_1)\times \cdots \times {{\,\textrm{SO}\,}}(k_p)$$ SO ( k 1 + ⋯ + k p ) / SO ( k 1 ) × ⋯ × SO ( k p ) and by imposing certain symmetry assumptions in the set of all left-invariant metrics on $${{\,\textrm{SO}\,}}(n)$$ SO ( n ) .