Constructing contact structures

This is a contact structure:

StandardContactStructure4 (by epsilon_is_afraid_of_zeta)

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 : 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.

Rod (by epsilon_is_afraid_of_zeta)

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.

xyplane (by epsilon_is_afraid_of_zeta) yzplane (by epsilon_is_afraid_of_zeta)

Standard Contact Structure (by epsilon_is_afraid_of_zeta)

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

Legendrian y-axis (by epsilon_is_afraid_of_zeta) Legendrian knot x-axis (by epsilon_is_afraid_of_zeta)

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.

radialfulldisk (by epsilon_is_afraid_of_zeta) Cylindrical Contact Structure (by epsilon_is_afraid_of_zeta)

cylindricalmodel (by epsilon_is_afraid_of_zeta)

~ by Ken Baker on April 6, 2008.

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