Abstract

Let (M,I) be an almost complex 6-manifold. The obstruction to the integrability of almost complex structure N : Λ^0,1(M) → Λ^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 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 Vol_I 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 Vol_I has an extremum in I if and only if (M,I) is nearly Kähler.

Keywords

nearly Kähler, Gray manifold, Hitchin functional, Calabi–Yau, almost complex structure

Mathematical Subject Classification

Primary: 53C15, 53C25

Authors