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 integrability of the almost complex structure (the so-called Nijenhuis tensor) N : Λ 0,1 (M ) −→ Λ 2,0 (M ) maps a 3-dimensional bundle to a 3-dimensional one. We say that Nijenhuis tensor is non-degenerate 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 non-degenerate Nijenhuis tensor is unique (up to a constant). We interpret the nearly Kähler property in terms of G 2 -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 non-degenerate 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.

Hodge theory on nearly Kähler manifolds

“…For a history of this notion, a number of equivalent definitions and a bibliography of current work in this field, we refer the reader to Mororianu, Nagy and Semmelmann [23] and our paper [31].…”

“…In [31] we showed that, unless a nearly Kähler manifold M is locally isometric to a 6-sphere, the almost complex structure on M is uniquely determined by the metric. Friedrich [9] proved this result for S 6 as well.…”

“…Friedrich [9] proved this result for S 6 as well. Also in [31] it was shown that the metric on M is uniquely determined by the almost complex structure.…”

Locally conformal calibrated $$G_2$$ G 2 -manifolds

“…In [6], a characterization of compact locally conformal parallel G 2 -manifolds as fiber bundles over S 1 with compact nearly Kähler fiber was obtained (see also [7]). …”

Homogeneous nearly Kähler manifolds

“…If there exists an almost complex structure J on M such that (M, g, J) is nearly Khler (non Khlerian) then it is unique. Moreover, in this case, J is invariant by the isometry group of g. This can be proved using the spinors (by [27], a 6-dimensional Riemannian manifold admits a nearly Khler structure if and only if it carries a real Killing spinor): see [8] proposition 2.4 and the reference therein [5], proposition 1, p126, or by a "cone argument" (see below) as in [43], proposition 4.7.…”