Classification des variété approximativement kähleriennes homogénes

Abstract

A Riemannian homogeneous manifold admitting a strict nearly-Kahler structure is 3-symmetric. We actually classify them in dimension 6 and use previous results of Swann, Cleyton and Nagy to prove the conjecture in higher dimensions. The six-dimensional homogeneous spaces, S3 × S3, S6, CP(3) and the flag manifold F(1, 2) have a unique (after a change of scale) nearly-Kahler, invariant structure. For the first one we solve a differential equation on the SU(3)-structure given by Reyes Carrion. For the last two it is obtained by canonical variation of the Kahler structure of the twistor space over a four-dimensional manifold. Finally, from Bar, a nearly-Kahler structure on the sphere S6 corresponds to a constant 3-form on the Riemannian cone R7.