[Paper Review] Locally conformally flat and self-dual structures on simple 4-manifolds

Mustafa Kalafat|arXiv (Cornell University)|Jan 28, 2013

Geometric Analysis and Curvature Flows11 references3 citations

TL;DR

This survey investigates the existence of locally conformally flat (LCF) and self-dual (SD) Riemannian metrics on simple 4-manifolds, such as products and simply-connected manifolds. Using curvature decomposition, the Gauss-Bonnet and signature theorems, and Kähler geometry, it proves that certain products like $\Sigma_g \times \Sigma_h$ with $g,h \geq 2$ cannot admit LCF or SD metrics due to topological obstructions, while others like $S^2 \times \Sigma_g$ for $g \geq 2$ do support such structures.

ABSTRACT

This is a survey article on the existence of locally conformally flat(LCF) and self-dual(SD) metrics on various basic 4-manifolds like simply-connected ones or product types

Motivation & Objective

• To systematically analyze the existence of locally conformally flat (LCF) and self-dual (SD) metrics on basic 4-manifolds, including product and simply-connected types.
• To unify and summarize scattered results on LCF and SD structures in 4-dimensional Riemannian geometry, particularly for Kähler and Einstein manifolds.
• To establish topological obstructions—especially via the signature and Euler characteristic—for the existence of LCF and SD metrics.
• To resolve open problems regarding the existence of such structures on products of surfaces of genus $g, h \geq 2$.
• To provide a comprehensive reference for researchers by consolidating tools and results from differential geometry and topology.

Proposed method

• Utilizes curvature decomposition into Weyl, Ricci, and scalar curvature components via the Kulkarni-Nomizu product and Bianchi identity.
• Applies the generalized Gauss-Bonnet theorem and Hirzebruch signature formula to relate curvature invariants to topological invariants $\chi(M)$ and $\tau(M)$.
• Employs the decomposition of the curvature tensor into $R = U \oplus Z \oplus W$, where $W$ is the Weyl tensor, and analyzes its self-dual and anti-self-dual parts $W_\pm$.
• Applies the characterization that a 4-manifold is LCF if and only if $W \equiv 0$, and SD if $W_- \equiv 0$, using Hodge star decomposition on 2-forms.
• Uses the fact that scalar-flat Kähler (SFK) surfaces are anti-self-dual, and applies this to product manifolds to deduce SD or LCF structure.
• Applies the Kähler-Einstein condition and the identity $|W_+|^2 = s^2/24$ to derive topological constraints via the Gauss-Bonnet formula.

Experimental results

Research questions

• RQ1 Which simply-connected or product-type 4-manifolds admit locally conformally flat (LCF) metrics?
• RQ2 Under what conditions does a Kähler 4-manifold admit a self-dual (SD) metric, and how does this relate to scalar-flatness?
• RQ3 What topological obstructions prevent a 4-manifold from admitting LCF or SD metrics?
• RQ4 Can the product of two hyperbolic surfaces $\Sigma_g \times \Sigma_h$ with $g,h \geq 2$ admit an LCF or SD metric?
• RQ5 What is the status of the existence of LCF metrics on $\Sigma_g \times \Sigma_h$ for $g \geq 2$, $h \geq 1$?

Key findings

• The product manifold $S^2 \times \Sigma_g$ with $g \geq 2$ admits a locally conformally flat metric, as its Weyl tensor vanishes due to the curvature lying in the image of the Kulkarni-Nomizu map.
• The product metric on $\Sigma_g \times \Sigma_h$ for $g,h \geq 2$ cannot be LCF or SD, as it would require $\chi = 0$, but $\chi = (2-2g)(2-2h) \neq 0$.
• The signature $\tau(\Sigma_g \times \Sigma_h) = 0$ due to the existence of an orientation-reversing diffeomorphism, but this does not suffice to make the metric SD or LCF.
• A Kähler-Einstein 4-manifold admitting an LCF metric must have Euler characteristic $\chi = 0$, as shown via the Gauss-Bonnet formula and the identity $|W_+|^2 = s^2/24$.
• The product $\Sigma_g \times \Sigma_h$ for $g,h \geq 2$ admits a Kähler-Einstein metric, but cannot admit an LCF metric due to the non-vanishing Euler characteristic.
• The existence of LCF metrics on $\Sigma_g \times \Sigma_h$ for $g \geq 2$, $h \geq 1$ remains an open problem, despite the topological obstruction being absent in some cases.