[Paper Review] Conformal equivalence between certain geometries in dimension 6 and 7

Richard Cleyton, Stefan Ivanov|ArXiv.org|Jul 20, 2006

Geometric Analysis and Curvature Flows13 references12 citations

TL;DR

This paper establishes that compact 6- and 7-dimensional manifolds in the conformal classes $Ω_1 + Ω_4$ (for SU(3) structures) and $Ψ_1 + Ψ_4$ (for G₂ structures) have constant scalar curvature if and only if they are either nearly parallel nearly Kähler or nearly parallel G₂ manifolds, respectively, or conformally equivalent to the standard sphere. The results rely on conformal equivalence to nearly parallel structures and curvature analysis using intrinsic torsion components and differential forms.

ABSTRACT

For G_2-manifolds the Fernández-Gray class X_1+X_4 is shown to consist of the union of the class X_4 of G_2-manifolds locally conformal to parallel G_2-structures and that of conformal transformations of nearly parallel or weak holonomy G_2-manifolds of type X_1. The analogous conclusion is obtained for Gray-Hervella class W_1+W_4 of real 6-dimensional almost Hermitian manifolds: this sort of geometry consists of locally conformally Kähler manifolds of class W_4 and conformal transformations of nearly Kähler manifolds in class W_1. A corollary of this is that a compact SU(3)-space in class W_1+W_4 or G_2-space of the kind X_1+X_4 has constant scalar curvature if only if it is either a standard sphere or a nearly parallel G_2 or nearly Kähler manifold, respectively. The properties of the Riemannian curvature of the spaces under consideration are also explored.

Motivation & Objective

• To characterize compact 6- and 7-dimensional manifolds in the conformal classes $Ω_1 + \u03a9_4$ and $\u03a8_1 + \u03a8_4$ with respect to scalar curvature.
• To prove that such manifolds have constant scalar curvature only if they are either nearly parallel or conformally equivalent to the standard sphere.
• To establish that the intrinsic torsion components $\mathcal{X}_1$ and $\mathcal{X}_4$ (or $\mathcal{W}_1$ and $\mathcal{W}_4$) generate the full class $\mathcal{X}_1 + \mathcal{X}_4$ (or $\mathcal{W}_1 + \mathcal{W}_4$) via conformal equivalence.
• To analyze the curvature structure of these manifolds, showing that Einstein metrics in the strict classes imply constant curvature and conformal equivalence to standard spheres.

Proposed method

• Use of differential forms $\omega_i$ whose exterior derivatives encode intrinsic torsion components for G₂ and SU(3) structures.
• Application of first-order identities derived from $d^2 = 0$ on these forms to constrain torsion components.
• Leveraging conformal invariance of the Weyl curvature to relate curvature forms across conformal classes.
• Utilization of Lemma 3 to show that non-vanishing functions satisfying $d\phi + \phi \alpha = 0$ imply $\alpha = -d\log|\phi|$.
• Proof via global conformal transformation arguments, showing that non-vanishing $\mathcal{X}_1$ or $\mathcal{W}_1$ components force global conformal equivalence to nearly parallel structures.
• Use of the first Bianchi identity and curvature decomposition to show that Einstein metrics in strict classes $\mathcal{X}_1 + \mathcal{X}_4$ or $\mathcal{W}_1 + \mathcal{W}_4$ must have constant curvature.

Experimental results

Research questions

• RQ1 Under what conditions does a compact 7-manifold in the $\mathcal{X}_1 + \mathcal{X}_4$ class have constant scalar curvature?
• RQ2 Is every compact 6-manifold in the $\mathcal{W}_1 + \mathcal{W}_4$ class with constant scalar curvature conformally equivalent to the standard 6-sphere or nearly Kähler?
• RQ3 Can the class $\mathcal{X}_1 + \mathcal{X}_4$ for $G_2$-structures be globally decomposed into conformal classes of $\mathcal{X}_4$ and $\mathcal{X}_1$?
• RQ4 What curvature properties characterize Einstein metrics in the strict classes $\mathcal{X}_1 + \mathcal{X}_4$ and $\mathcal{W}_1 + \mathcal{W}_4$?
• RQ5 Does the existence of non-vanishing $\mathcal{X}_1$ or $\mathcal{W}_1$ torsion component imply global conformal equivalence to a nearly parallel structure?

Key findings

• A compact 7-dimensional $G_2$-manifold in class $\mathcal{X}_1 + \mathcal{X}_4$ has constant scalar curvature if and only if it is either nearly parallel or conformally equivalent to the standard 7-sphere.
• A compact 6-dimensional almost Hermitian manifold in class $\mathcal{W}_1 + \mathcal{W}_4$ has constant scalar curvature if and only if it is either nearly Kähler or conformally equivalent to the standard 6-sphere.
• The class $\mathcal{X}_1 + \mathcal{X}_4$ for $G_2$-structures is globally generated by $\mathcal{X}_1$ (nearly parallel) and $\mathcal{X}_4$ (locally conformally parallel), with non-vanishing $\mathcal{X}_1$ implying global conformal equivalence.
• For complete Einstein metrics in the strict class $\mathcal{X}_1 + \mathcal{X}_4$, the manifold is isometric to a space of constant curvature: sphere, hyperbolic space, or Euclidean space.
• The curvature tensor of such manifolds decomposes as $R^g = R^{\mathfrak{g}} + \frac{s_g}{2n(n-1)}\mathrm{Id}_{\Lambda^2}$, with $R^{\mathfrak{g}}$ corresponding to the nearly parallel holonomy algebra.
• If the curvature is of the form (7.1), then $R^{\mathfrak{g}} = 0$, implying the manifold has constant sectional curvature and $R^{\mathfrak{g}} = 0$.