Characteristic Classes for the Degenerations of Two-Plane Fields in Four Dimensions

Details Characteristic Classes for the Degenerations of Two-Plane Fields in Four Dimensions Characteristic Classes for the Degenerations of Two-Plane Fields in Four Dimensions

1997-04-02

arxiv.org/abs/dg-ga/9704001v1

Mathematics

Differential Geometry

LaTeX, 15 pages

Scientific paper

There is a remarkable type of field of two-planes special to four dimensions known as an Engel distributions. They are the only stable regular distributions besides the contact, quasi-contact and line fields. If an arbitrary two-plane field on a four-manifold is slightly perturbed then it will be Engel at generic points. On the other hand, if a manifold admits an oriented Engel structure then the manifold must be parallelizable and consequently the alleged Engel distribution must have a degeneration loci -- a point set where the Engel conditions fails. By a theorem of Zhitomirskii this locus is a finite union of surfaces. We prove that these surfaces represent Chern classes associated to the distribution.

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