Constructing contact structures

This is a contact structure:

This is too.

I’ve shown other pictures of these things, but what are they?

A 2-plane field ξ on a smooth 3-manifold M may be thought of as the kernel of a 1-form or as a smooth choice of an R² subspace within the tangent space T_p M at each point p of M. At each point p of M think of a plane with its origin at that point. A 2-plane field on M is integrable if at each point p of M there is a small open chunk of a surface in M containing that point for which the tangent planes to the surface coincide with those of the 2-plane field. A 2-plane field on M is a contact structure if it is nowhere integrable.

If our 2-plane field ξ is the kernel of the 1-form a then:

• it is integrable if α^dα=0 at each point and
• it is a (positive) contact structure if α^dα>0 at each point.

There are two common models for a contact structure in R³: Cartesian and cylindrical. Both begin with a ray or line of planes that, looking down the line, rotate clockwise as they come towards us and counterclockwise as they go away.

Cartesian model

Make an array of these rods sweeping them left-right and up-down, eventually filling space. This may be taken to be the kernel of the 1-form α=dz-ydx.

Look down the y-axis and the x-axis.

Cylindrical model

Start off with our rod of planes horizontal at the center, sweep it around in a circle, and then sweep it up-down.

~ by Ken Baker on April 6, 2008.