[Paper Review] Transverse conformal Killing forms and a Gallot-Meyer Theorem for foliations

Seoung Dal Jung, Ken Richardson|ArXiv.org|May 27, 2008

Geometric Analysis and Curvature Flows4 references4 citations

TL;DR

This paper establishes a Gallot-Meyer-type spectral inequality for transverse conformal Killing forms on Riemannian foliations with $C$-positive normal curvature. It proves that if a closed basic 1-form satisfies $\Delta_B\phi = qC\phi$, then the foliation is transversally isometric to a quotient of the $q$-sphere, extending classical Obata-type rigidity results to the foliated setting using transverse differential geometry and basic Laplacian analysis.

ABSTRACT

We study transverse conformal Killing forms on foliations and prove a Gallot-Meyer theorem for foliations. Moreover, we show that on a foliation with $C$-positive normal curvature, if there is a closed basic 1-form $ϕ$ such that $Δ_Bϕ=qCϕ$, then the foliation is transversally isometric to the quotient of a $q$-sphere.

Motivation & Objective

• To extend the Gallot-Meyer theorem to the setting of Riemannian foliations with $C$-positive normal curvature.
• To study transverse conformal Killing forms and their spectral properties on foliated manifolds.
• To establish a rigidity result linking eigenvalues of the basic Laplacian to transverse geometry.
• To characterize when a foliation is transversally isometric to a sphere via spectral conditions on basic forms.

Proposed method

• Define transverse conformal Killing forms as basic differential forms satisfying a twisted first-order equation involving the transverse Levi-Civita connection and the twistor operator.
• Use the basic Laplacian $\Delta_B$ acting on basic forms to analyze spectral gaps under curvature assumptions.
• Apply a modified version of Meyer’s theorem to derive lower bounds on eigenvalues of $\Delta_B$ depending on whether $d_B\phi=0$ or $\delta_B\phi=0$.
• Utilize the tautness theorem to assume vanishing mean curvature $\kappa=0$ without loss of generality, simplifying curvature estimates.
• Prove that if $\Delta_B\phi = qC\phi$ for a closed basic 1-form $\phi$, then the foliation is transversally isometric to a quotient of the $q$-sphere.
• Use the generalized Obata theorem for foliations to link spectral data to global transverse geometry.

Experimental results

Research questions

• RQ1 Under what conditions on the transverse curvature does the basic Laplacian on a foliation admit a spectral gap?
• RQ2 Can a Gallot-Meyer-type eigenvalue estimate be established for transverse conformal Killing forms on foliated manifolds?
• RQ3 What geometric structure arises when a closed basic 1-form achieves the minimal eigenvalue $qC$ under $C$-positive normal curvature?
• RQ4 How do special transverse Killing forms with constant $\beta$ relate to curvature and spectral properties?
• RQ5 Is there a transverse analog of the Obata rigidity theorem for foliations?

Key findings

• For any basic $r$-form $\phi$ with $1 \leq r \leq q-1$, the eigenvalue $\lambda_B$ of the basic Laplacian satisfies $\lambda_B \geq r(q-r+1)C$ if $d_B\phi = 0$, and $\lambda_B \geq (r+1)(q-r)C$ if $\delta_B\phi = 0$.
• If a closed basic 1-form $\phi$ satisfies $\Delta_B\phi = qC\phi$ and the foliation has $C$-positive normal curvature, then the foliation is transversally isometric to the quotient of the $q$-sphere by a finite subgroup of $O(q)$.
• Special transverse Killing $r$-forms with constant $\beta$ satisfy $\beta \leq -(r+1)C$, with equality holding when the transversal sectional curvature is constant $C$.
• When the transversal curvature is constant $C$, all transverse Killing $r$-forms are special with $\beta = -(r+1)C$.
• The basic Laplacian preserves the decomposition of $L^2$-forms into basic and non-basic components, and the spectral estimates are invariant under metric modifications that preserve the transverse geometry and $\kappa=0$.
• The proof relies on modifying the bundle-like metric to eliminate the mean curvature form while preserving the basic Laplacian and transverse metric, enabling application of curvature estimates.