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 SO(n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\ extrm{SO}\\,}}(n)$$\\end{document} which are not naturally reductive. This is achieved by using the real flag manifolds SO(k1+⋯+kp)/SO(k1)×⋯×SO(kp)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\ extrm{SO}\\,}}(k_1+\\cdots +k_p)/{{\\,\ extrm{SO}\\,}}(k_1)\ imes \\cdots \ imes {{\\,\ extrm{SO}\\,}}(k_p)$$\\end{document} and by imposing certain symmetry assumptions in the set of all left-invariant metrics on SO(n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\ extrm{SO}\\,}}(n)$$\\end{document}.