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

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 $\varphi$ such that ${\mathrm{\Delta }}_B\varphi =qC\varphi$, then the foliation is transversally isometric to the quotient of a $q$-sphere.