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: Λ0,1 (M) → Λ2,0 (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 Kähler if it admits a Hermitian form ω such that ∇(ω) is totally antisymmetric, ∇ being the Levi-Civita connection. We show that a nearly Kähler metric on a given almost complex 6-manifold with nondegenerate Nijenhuis tensor is unique (up to a constant). We interpret the nearly Kähler property in terms of G2- geometry and in terms of connections with totally antisymmetric torsion, obtaining a number of equivalent definitions. We construct a natural diffeomorphism-invariant functional I → ∫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 → ∫M VolI has an extremum in I if and only if (M, I) is nearly Kähler.