[Paper Review] G2-structure deformations and warped products

Sergey Grigorian|arXiv (Cornell University)|Oct 20, 2011

Geometric and Algebraic Topology10 references3 citations

TL;DR

This paper investigates non-infinitesimal deformations of G₂-structures on 7-manifolds, focusing on vector field-induced deformations in the 7-dimensional representation of G₂. It proves that transitions from conformally nearly parallel G₂-structures (torsion in W₁⊕W₇) to nearly parallel G₂-structures (torsion in W₁) exist if and only if the metric is a specific warped product, generalizing constructions by Cleyton and Ivanov.

ABSTRACT

We overview the properties of non-infinitesimal deformations of G2-structures on seven-manifolds, and in particular, focus on deformations that lie in the seven-dimensional representation of G2 and are thus defined by a vector. We then consider deformations from G2-structures with the torsion class having one-dimensional and seven-dimensional components (so-called conformally nearly parallel G2-manifolds) to G2-structures with just a one-dimensional torsion component (nearly parallel G2-manifolds). We find that deformations between such structures exist if and only if the metric is a particular warped product metric.

Motivation & Objective

• To understand non-infinitesimal deformations of G₂-structures on 7-manifolds, particularly those induced by vector fields.
• To analyze the conditions under which a G₂-structure with torsion in W₁⊕W₇ can be deformed into one with torsion in W₁.
• To characterize the metric structure required for such deformations to exist, linking to warped product geometry.
• To extend previous results on torsion classes and deformations in G₂-geometry, particularly in the context of M-theory compactifications.

Proposed method

• The paper uses the general deformation framework from prior work (Grigorian, 2010) to compute the new torsion after a deformation in the 7-dimensional space Λ₇³ of 3-forms.
• It applies the torsion transformation equations derived in [10] to the specific case of deforming from W₁⊕W₇ to W₁ torsion classes.
• The analysis involves expressing the Hessian of the warp factor f in terms of the torsion components τ₁ and τ₇, leading to a second-order PDE on f.
• By solving a differential equation for a function F(f) such that ∇∇F is proportional to the metric, the method identifies conditions under which the metric must be a warped product.
• The derivation uses the conformal invariance of the W₁ torsion class and relates the deformation to the existence of a warped product metric.
• It connects the results to known constructions by Cleyton and Ivanov, showing consistency with their warped product metrics over nearly Kähler 6-manifolds.

Experimental results

Research questions

• RQ1 Under what conditions can a G₂-structure with torsion in W₁⊕W₇ be deformed into one with torsion in W₁ via a non-infinitesimal deformation in Λ₇³?
• RQ2 What metric structure is required for such a deformation to exist, and how does it relate to warped product geometry?
• RQ3 How do the torsion components τ₁ and τ₇ evolve under such deformations, and what ODEs govern their behavior?
• RQ4 Can the deformation process be reversed or extended to other torsion classes, particularly W₂₇?
• RQ5 What is the role of conformal transformations in relating nearly parallel and conformally nearly parallel G₂-structures?

Key findings

• A deformation from a G₂-structure with torsion in W₁⊕W₇ to one with torsion in W₁ exists if and only if the metric is a warped product of the form g = |∇F|⁻² dF² + |∇F|² ̂g for a specific function F derived from the warp factor f.
• The Hessian of F satisfies ∇ₐ∇ᵦF = P(f)gₐᵦ + Q(f)∇ₐf∇ᵦf, with Q(f) = 2(f² - 3f - 6)/(f(f² - 9)), which leads to the warped product condition.
• The function F is determined by solving d²F/df² + (dF/df)Q(f) = 0, yielding G = dF/df = 6(f−3)²ᐟ³ / (f⁴ᐟ³(f+3)⁴ᐟ³), which integrates to give the warped product structure.
• The resulting metric matches the warped product construction of Cleyton and Ivanov, where the metric is dt² + h(t)² ̂g with h(t) satisfying h' = h cosθ − σ sinθ and θ' = σ sinθ / h.
• The torsion components are explicitly given by τ₁ = h⁻¹σ sinθ and τ₇ = h⁻¹(σ cosθ − h')dt, consistent with the W₁⊕W₇ class.
• Solutions exist for appropriate initial conditions on h and θ, and the deformation process preserves the geometric structure under the specified metric condition.