Bismut Ricci flat generalized metrics on compact homogeneous spaces

A generalized metric on a manifold M M , i.e., pair alttext="left-parenthesis g comma upper H right-parenthesis"> ( g , H stretchy="false">) encoding="application/x-tex">(g,H) where alttext="g"> encoding="application/x-tex">g is Riemannian and H"> encoding="application/x-tex">H closed alttext="3"> 3 encoding="application/x-tex">3 -form, fixed point of the Ricci flow if only Bismut flat: -harmonic R c left-parenthesis right-parenthesis equals one fourth Subscript Superscript 2"> R c = 1 4 2 encoding="application/x-tex">Rc(g)=\tfrac {1}{4}H_g^2 . On any homogeneous space M G slash K"> G / K encoding="application/x-tex">M=G/K 1 times × encoding="application/x-tex">G=G_1\times G_2 compact semisimple Lie group with two simple factors, under some mild assumptions, we exhibit flat G"> encoding="application/x-tex">G -invariant metric, which proved to be unique among alttext="4"> encoding="application/x-tex">4 -parameter metrics in many cases, including when encoding="application/x-tex">K neither abelian nor semisimple. other hand, standard Einstein both pi K π encoding="application/x-tex">G_1/\pi _1(K) 2 encoding="application/x-tex">G_2/\pi _2(K) give one-parameter family encoding="application/x-tex">G/K show that it most likely pairwise non-homothetic by computing ratio eigenvalues. This case for every form normal Delta Δ<!-- Δ encoding="application/x-tex">M=G\times G/\Delta K 35 Baseline S O 8 7 35 S O 8 7 encoding="application/x-tex">M^{35}=SO(8)\times SO(7)/G_2