Recall a tranverse metric on a (regular) foliated manifold $(M,F)$ is a positive symmetric $C^\infty (M)$-bilinear form $g$ such that
1) $Ker(g_x)=T_x F$
I know that not every foliation $(M,F)$ admits such a tranverse metric, however, I would like to know some simple examples of when this fails. I do know that if the foliation arises as the fibers of a sumbersion, then it always admits a transverse metric, however I would also like to know some examples of foliations not of this form which DO admit a tranverse metric. Thank you!