Chern classes of integral submanifolds of some contact manifolds

A complex subbundle of the normal bundle to an integral submanifold of the contact distribution in a Sasakian manifold is given. The geometry of this bundle is investigated and some results concerning its Chern classes are obtained.

On the rank of horizontal maps

Let M, N be smooth manifolds and ℋ ⊂ TN a smooth sub-bundle. A smooth map φ:M → N will be called horizontal ifAt points where φ(M) is a submanifold of N, the tangent spaces to φ(M) will be sub-spaces of the fibres of ℋ. In other words where φ(M) is a submanifold it is an integral submanifold of ℋ. If ℋ is not integrable it will follow that the rank of Æ must be less than dim ℋx. But we can often place much stricter bounds on the rank of φ by examining the integrability tensor of ℋ. This we shall do in this note.