[Paper Review] Flat nearly Kähler manifolds

Vicente Cortés, Lars Schaefer|ArXiv.org|Oct 5, 2006

Geometry and complex manifolds6 references4 citations

TL;DR

This paper classifies flat strict nearly Kähler manifolds with indefinite metrics, showing they are locally products of a flat pseudo-Kähler factor and a strict nearly Kähler manifold of split signature (2m,2m) with m ≥ 3. The geometry of the latter is encoded in a complex three-form ζ ∈ Λ³(ℂᵐ)⁎, and the first nontrivial example arises in dimension 12.

ABSTRACT

We classify flat strict nearly Kähler manifolds with (necessarily) indefinite metric. Any such manifold is locally the product of a flat pseudo-Kähler factor of maximal dimension and a strict flat nearly Kähler manifold of split signature $(2m,2m)$ with $m\ge 3$. Moreover, the geometry of the second factor is encoded in a complex three-form $ζ\in Λ^3 (\mathbb{C}^m)^*$. The first nontrivial example occurs in dimension $4m=12$.

Motivation & Objective

• To classify flat strict nearly Kähler manifolds equipped with indefinite (pseudo-Riemannian) metrics.
• To understand the geometric structure of such manifolds, particularly their local decomposition and holonomy properties.
• To establish a correspondence between complete simply connected flat nearly Kähler manifolds without pseudo-Kähler factors and GLₘ(ℂ)-orbits on complex three-forms with maximal support.
• To clarify the role of the canonical connection with skew-symmetric torsion in the nearly Kähler condition under flatness.
• To show the non-existence of such manifolds with definite metrics, except in the Kähler case.

Proposed method

• Use the flatness of the Levi-Civita connection to construct a canonical Hermitian connection ∇ with totally skew-symmetric torsion T = -2η, where η = ½JDJ.
• Characterize nearly Kähler structures via a real three-form η ∈ Λ³V satisfying two constraints: isotropic support and a type condition.
• Represent the three-form η as η = ζ + ζ̄, where ζ ∈ Λ³(ℂᵐ)⁎ is a complex three-form, encoding the geometry of the (2m,2m)-signature factor.
• Prove that the isotropic support of η is a J_can-invariant complex subspace L ⊂ ℂᵐᵐ of complex dimension m ≥ 3.
• Apply a decomposition theorem to show that any strict flat nearly Kähler manifold locally splits as a product of a flat pseudo-Kähler factor and a strict (2m,2m)-signature factor.
• Establish a bijective correspondence between GLₘ(ℂ)-orbits on the open subset of three-forms with maximal support and isomorphism classes of complete simply connected nearly Kähler manifolds without pseudo-Kähler de Rham factors.

Experimental results

Research questions

• RQ1 What is the local geometric structure of flat strict nearly Kähler manifolds with indefinite metrics?
• RQ2 Which real three-forms η ∈ Λ³V give rise to flat nearly Kähler structures, and what constraints must they satisfy?
• RQ3 Can a complete simply connected flat nearly Kähler manifold without pseudo-Kähler de Rham factor be constructed from a complex three-form ζ ∈ Λ³(ℂᵐ)⁎?
• RQ4 Why do such manifolds only exist in dimensions 4m ≥ 12, and why are definite-metric examples excluded?
• RQ5 How do GLₘ(ℂ)-orbits on Λ³(ℂᵐ)⁎ classify isomorphism types of these manifolds?

Key findings

• Any flat strict nearly Kähler manifold with indefinite metric is locally isomorphic to a product of a flat pseudo-Kähler manifold of maximal dimension and a strict nearly Kähler manifold of split signature (2m,2m) with m ≥ 3.
• The geometry of the (2m,2m)-signature factor is encoded in a complex three-form ζ ∈ Λ³(ℂᵐ)⁎, with η = ζ + ζ̄ determining the structure.
• The first nontrivial example of a strict flat nearly Kähler manifold occurs in dimension 12, corresponding to m = 3.
• There are no strict flat nearly Kähler manifolds with definite (positive or negative) metrics, as such manifolds must be Kähler.
• A complete simply connected flat nearly Kähler manifold without pseudo-Kähler de Rham factor exists if and only if the associated three-form ζ has maximal support.
• The isomorphism classes of such manifolds are in bijective correspondence with GLₘ(ℂ)-orbits on the open subset of Λ³(ℂᵐ)⁎ consisting of three-forms with maximal support.