[Paper Review] An intrinsic volume functional on almost complex 6-manifolds and nearly Kaehler geometry

Misha Verbitsky|ArXiv.org|Jul 8, 2005

Geometry and complex manifolds11 references4 citations

TL;DR

This paper introduces an intrinsic volume functional on almost complex 6-manifolds with non-degenerate Nijenhuis tensor and proves that its extrema correspond precisely to nearly Kähler structures. It establishes that a nearly Kähler metric on such a manifold is unique up to scaling, and characterizes nearly Kähler geometry via G₂-structures on the metric cone and connections with totally antisymmetric torsion.

ABSTRACT

Let $(M,I)$ be an almost complex 6-manifold. The obstruction to integrability of almost complex structure (so-called Nijenhuis tensor) 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 is called nearly Kaehler if it admits a Hermitian form $ω$ such that $ abla(ω)$ is totally antisymmetric, $ abla$ being the Levi-Civita connection. We show that a nearly Kaehler metric on a given almost complex 6-manifold with non-degenerate Nijenhuis tensor is unique (up to a constant). We interpret the nearly Kaehler property in terms of G_2-geometry and in terms of connections with totally antisymmetric torsion, obtaining a number of equivalent definitions. Further on, we construct an intrinsic diffeomorphism-invariant functional 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 intrinsic volume functional has an extremum in $I$ if and only if $(M,I)$ is nearly Kaehler.

Motivation & Objective

• To define a diffeomorphism-invariant volume functional on the space of almost complex structures on a 6-manifold, analogous to Hitchin's functional.
• To investigate the critical points of this functional in the context of almost complex structures with non-degenerate Nijenhuis tensor.
• To characterize nearly Kähler structures on 6-manifolds through geometric and analytic conditions, including connections with totally antisymmetric torsion.
• To establish the uniqueness of nearly Kähler metrics on 6-manifolds with non-degenerate Nijenhuis tensor, up to a constant scaling.
• To unify different definitions of nearly Kähler geometry via G₂-structures on the Riemannian cone and torsionful Hermitian connections.

Proposed method

• Define the intrinsic volume functional $ \Psi(I) = \int_M \operatorname{Vol}_I $, where $ \operatorname{Vol}_I $ is derived from the determinant of the Nijenhuis tensor and its dual.
• Use the non-degeneracy of the Nijenhuis tensor to identify $ \Lambda^{1,0}(M) \cong \Lambda^{2,0}(M) $ via a canonical 3-form, enabling tensorial identifications.
• Apply infinitesimal variation theory for almost complex structures, parameterized by $ \delta \in \Lambda^{0,1}(M) \otimes T^{1,0}(M) $, to compute the first variation of $ \Psi $.
• Derive the first variation formula $ \frac{d\Psi}{dI}(\delta) = 2\operatorname{Re} \int_M \overline{\partial}\delta \wedge \omega $, where $ \omega $ is the Hermitian form.
• Use integration by parts to show that $ \Psi $ has an extremum iff $ \overline{\partial}\omega = 0 $, or equivalently $ d\omega \in \Lambda^{3,0} \oplus \Lambda^{0,3} $.
• Establish equivalence between nearly Kähler geometry and the condition $ \nabla\omega $ totally antisymmetric, linking it to $ G_2 $-structures on the cone over $ M $.

Experimental results

Research questions

• RQ1 What conditions on an almost complex 6-manifold with non-degenerate Nijenhuis tensor ensure that the intrinsic volume functional $ \Psi(I) = \int_M \operatorname{Vol}_I $ achieves an extremum?
• RQ2 How is the nearly Kähler condition characterized in terms of connections with totally antisymmetric torsion and Hermitian geometry?
• RQ3 What is the relationship between nearly Kähler structures on 6-manifolds and $ G_2 $-structures on their Riemannian cones?
• RQ4 Is the nearly Kähler metric on a 6-manifold with non-degenerate Nijenhuis tensor uniquely determined by the almost complex structure?
• RQ5 Under what conditions does the existence of a Hermitian connection with totally antisymmetric torsion imply that the almost complex structure is nearly Kähler?

Key findings

• The intrinsic volume functional $ \Psi(I) = \int_M \operatorname{Vol}_I $ has an extremum at an almost complex structure $ I $ if and only if $ d\omega \in \Lambda^{3,0}(M) \oplus \Lambda^{0,3}(M) $, where $ \omega $ is the Hermitian form.
• An almost complex 6-manifold with non-degenerate Nijenhuis tensor admits a Hermitian connection with totally antisymmetric torsion if and only if it is nearly Kähler.
• The nearly Kähler metric on such a manifold is uniquely determined by the almost complex structure up to a constant scaling factor.
• Nearly Kähler structures on 6-manifolds are equivalent to $ G_2 $-structures on the Riemannian cone over the manifold, providing a geometric characterization.
• The nearly Kähler condition is equivalent to the requirement that $ \nabla\omega $ is totally antisymmetric, where $ \nabla $ is the Levi-Civita connection.
• The extremum condition for the volume functional is equivalent to the vanishing of $ \overline{\partial}\omega $, which is equivalent to the nearly Kähler condition under the non-degeneracy assumption on the Nijenhuis tensor.