An intrinsic volume functional on almost complex 6-manifolds and nearly Kähler geometry

Abstract

Let (M, I) be an almost complex 6-manifold. The obstruction to the integrability of almost complex structure N:3 0,1 (M)!3 2,0 (M) (the so-called Nijenhuis tensor) maps one 3-dimensional bundle to another 3-dimensional bundle. We say that Nijenhuis tensor is nondegenerate if it is an isomorphism. An almost complex manifold (M, I) is called nearly Kahler if it admits a Hermitian form ! such that r(!) is totally antisymmetric, r being the Levi-Civita connection. We show that a nearly Kahler metric on a given almost complex 6-manifold with nondegenerate Nijenhuis tensor is unique (up to a constant). We interpret the nearly Kahler property in terms of G2geometry and in terms of connections with totally antisymmetric torsion, obtaining a number of equivalent definitions. Weconstructanaturaldiffeomorphism-invariantfunctional I ! R M VolI on the space of almost complex structures on M, similar to the Hitchin functional, and compute its extrema in the following important case. Consider an almost complex structure I with nondegenerate Nijenhuis tensor, admitting a Hermitian connection with totally antisymmetric torsion. We show that the Hitchin-like functional I ! R M VolI has an extremum in I if and only if (M, I) is nearly Kahler.