Transverse conformal Killing forms and a Gallot-Meyer Theorem for foliations

TL;DR

This paper investigates transverse conformal Killing forms on foliations, proving a Gallot-Meyer theorem and characterizing foliations with positive normal curvature via spectral conditions, leading to a transversally spherical structure.

Contribution

It establishes a Gallot-Meyer type theorem for foliations and characterizes foliations with positive normal curvature using spectral properties of basic forms.

Findings

Proves a Gallot-Meyer theorem for foliations.

Shows that certain spectral conditions imply the foliation is transversally spherical.

Characterizes foliations with positive normal curvature via basic form eigenvalues.

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 $\phi$ such that $\Delta_B\phi=qC\phi$, then the foliation is transversally isometric to the quotient of a $q$-sphere.