  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.
