BISMUT RICCI FLAT GENERALIZED METRICS ON COMPACT HOMOGENEOUS SPACES

A generalized metric on a manifold M, i.e., a pair (g, H), where gis a Riemannian metric and H a closed 3-form, is a fixed point of the generalized Ricci flow if and only if (g, H) is Bismut Ricci flat: H is g-harmonic and Rc(g) = 14Hg2. On any homogeneous space M = G/K, where G = G1 x G2 is a compact semisimple Lie group with two simple factors, under some mild assumptions, we exhibit a Bismut Ricci flat G-invariant generalized metric, which is proved to be unique among a 4-parameter space of metrics in many cases, including when K is neither abelian nor semisimple. On the other hand, if K is simple and the standard metric is Einstein on both G1/& pi;1(K) and G2/& pi;2(K), we give a one-parameter family of Bismut Ricci flat G-invariant generalized metrics on G/K and show that it is most likely pairwise non-homothetic by computing the ratio of Ricci eigenvalues. This is proved to be the case for every space of the form M = G x G/& UDelta;K and for M35 = SO(8) x SO(7)/G2.