Infinitesimal Einstein deformations of nearly Kähler metrics

Abstract:

It is well known that every 6-dimensional strictly nearly Kähler manifold $(M,g,J)$ is Einstein with positive scalar curvature $\operatorname {scal}>0$. Moreover, one can show that the space $E$ of co-closed primitive $(1,1)$-forms on $M$ is stable under the Laplace operator $\Delta$. Let $E(\lambda )$ denote the $\lambda$-eigenspace of the restriction of $\Delta$ to $E$. If $M$ is compact, and has normalized scalar curvature $\operatorname {scal}=30$, we prove that the moduli space of infinitesimal Einstein deformations of the nearly Kähler metric $g$ is naturally isomorphic to the direct sum $E(2)\oplus E(6)\oplus E(12)$. From Moroianu, Nagy, and Semmelmann (2008), the last summand is itself isomorphic with the moduli space of infinitesimal nearly Kähler deformations.