A note on projective foliations

Abstract

In [12], Nishikawa and Sato studied conformal and projective foliationsdefined as foliations whose second order transversal bundle is endowed witheither a conformal or a projective projectable structure. (See, for instance[7] for such structures on manifolds.) Namely, they proved the existence ofcorresponding projectable normal Cartan connections from which they deduce thatthe same strong Bott vanishing phenomenon like for Riemannian foliations holds.Then, Nishikawa studied characteristic classes of projective foliations in [13].Independently, I discussed conformal foliations in [16] using the classicaldefinition of conformal structures by means of Riemannian metrics, and Montesinos[11] proved the strong Bott vanishing theorem for conformal foliations, by usingthis classical approach.The aim of this Note is to present projective foliations by using the ,alternate approach to projective structures known as geometry of paths [6], andby constructing the normal connection with a vector bundle version of the originalCartan technique [4] . This approach will provide us not only with the Bott-Nishikawa-Sato vanishing theorem of [12, 13] but also with projectively invariantrepresentative forms of the real Pontrjagin classes of manifolds and of transversebundles of foliations. Furthermore, we shall obtain a cohomological obstructionto the existence of a transversal projective projectable (I am using the narrefoliate instead) structure.Beyond all this, since two approaches to projective structures on manifoldsare available, it seems natural to use them both in studying projective foliationsas well.109