Linear and Convex Tori

Here’s a picture of what’s basically a neighborhood of a constant radius torus in the cylindrical model of a contact structure — mod out by the z-action, of course.  Okay, so I’ve reembedded this… (you could look at the xz-plane modded out by a lattice too).

The intersection of the contact planes with a surface in a contact manifold gives a (typically singular) line field on the surface.  This line field integrates to give the characteristic foliation.

In this model, intersecting the contact planes with the surface suggests the line field and hence the characteristic foliation.  In the present case, this is quite literal.

The reembedding was chosen so that the characteristic foliation is a bunch of vertical parallel circles.  Nearby constant radius tori will also have characteristic foliations by circles or lines according to the rationality of the slope of the line field.  I tend to call thusly situated tori linear.

Linear tori have their place in the theory of contact structures, but this linear ability is special to surfaces of euler characteristic 0.  The notion of convexity however applies to all surfaces.

A surface in a contact manifold is convex if in some neighborhood of the surface there is a transverse flow that preserves the contact structure.  Think about translating the xy-plane of the standard euclidean model in the z direction — or think about a constant radius neighborhood of the x-axis in this model. The motivating thing is that every surface is $C^\infty$ close to a convex surface. (That’s not quite the whole story for a surface with boundary.)

So let us perturb our linear torus into a convex torus. We may do so by picking two circles in the characteristic foliation of the linear torus and pushing the interiors of the complementary annuli to opposite sides of the torus.

Intersecting with the contact planes suggests the characteristic foliation.

This characteristic foliation has only two closed circle leaves and the rest are lines. The contact preserving transverse flow is a bit harder to see here… it’s not just expansion in the radial direction. For instance, along the two closed leaves it will have to be tangent to the former linear torus. (We’ll save this for another time.)

There are many ways of perturbing a surface into a convex one…