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 →∫MVolI 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 →∫MVolI 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
