Click here to flash read.
A generalized metric on a manifold $M$, i.e., a pair $(g,H)$, where $g$ is 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 $ric(g)=\tfrac{1}{4} H_g^2$. On any homogeneous space $M=G/K$,
where $G=G_1\times G_2$ 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 $G_1/\pi_1(K)$ and $G_2/\pi_2(K)$, 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 $M=G\times G/\Delta K$ and for $M^{35}=SO(8)\times SO(7)/G_2$.
A Corrigendum has been added in Appendix A.
No creative common's license