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Contact geometry

From Academic Kids

In mathematics, contact geometry is the study of completely nonintegrable hyperplane fields on manifolds. From the Frobenius theorem, one recognizes that this is (roughly) the opposite of a foliation. As its sister, symplectic geometry, belongs to the even-dimensional world, contact geometry is the odd-dimensional counterpart.

Contact geometry has — as does symplectic geometry — broad applications in physics, e.g. geometrical optics, classical mechanics, thermodynamics, geometric quantization, and applied mathematics such as control theory. One can prove amusing things, like 'You can always parallel-park your car, provided the space is big enough'. Mathematical research reveals many exciting applications especially in low-dimensional topology.

A contact form α on a 2n+1 dimensional manifold M is a (local) 1-form with the property that

<math> \alpha \wedge (d\alpha)^n\ne 0. <math>

A contact structure ξ on a manifold is the kernel of a contact form α, i.e. a completely nonintegrable hyperplane field. Roughly this means that you cannot find a piece of a hypersurface tangent to ξ on an open set.

It also follows from this definition that dα, when restricted to ξ, is nondegenerate. This means that ξ is a symplectic bundle on the manifold. Since symplectic spaces are even-dimensional, contact manifolds need to be odd dimensional.

As a prime example, consider on R3, endowed with coordinates

(x, y, z),

the 1-form

dz -ydx.

The contact plane ξ at a point

(x,y,z)

is spanned by vectors

X1 = ∂y

and

X2 = ∂x+y∂z.

(Draw a picture of this!). Actually one can generalize this example to any R2n+1. By a theorem of Darboux, every contact structure on a manifold looks locally like this!

The cotangent bundle T* M of any n-dimensional manifold M is itself a manifold (of dimension 2n) and supports naturally an exact symplectic structure ω = dλ. (This 1-form λ is sometimes called Liouville form). Choose a Riemannian metric on the manifold. That allows one to consider the unit sphere in each cotangent plane. The Liouville form restricted to the unit cotangent bundle is a contact structure. The vector field A (uniquely) defined by λ(A)=1 and dλ(A, B)=0 for all vector fields B generates the geodesic flow of this metric.

On the other hand, one can build a contact manifold by considering the manifold T*M× R. With coordinates (x,t) this has a contact structure

α=dt+λ.

The last example showed how to obtain contact manifolds from symplectic ones. Vice versa one gets a symplectic manifold out of a contact manifold by crossing with R: If α is a contact form for a manifold M, then

ω=d(etα)

is a symplectic form on M×R, where t denotes the variable in the R-direction.

For a good starting point for further readings consult:

  • Etnyre, J. Introductory lectures on contact geometry, Proc. Sympos. Pure Math. 71 (2003), 81-107.arXiv (http://front.math.ucdavis.edu/math.SG/0111118)
  • Geiges, H. Contact Geometry, arXiv (http://front.math.ucdavis.edu/math.SG/0307242)
  • Aebischer et.al. symplectic geometry, Birkhäuser, 1994.

Some historical remarks

The threads to the beginning of contact geometry may lead back to the work of Christiaan Huygens, Barrow (http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Barrow.html) and Isaac Newton, i.e. to the beginning of modern mathematics. The theory of contact transformations (i.e. transformations preserving a contact structure) was developed by Sophus Lie, with two starting points: once as a tool to study differential equations (e.g. Legendre transformation) and second to describe the 'change of space element', familiar in projective duality.

For further information on the history of contact geometry, see

  • Lutz, R. Quelques remarques historiques et prospectives sur la géométrie de contact , Conf. on Diff.Geom. and Top. (Sardinia, 1988) Rend. Fac. Sci. Univ. Cagliari 58 (1988), suppl., 361-393.
  • Geiges, H. A Brief History of Contact Geometry and Topology, Expo. Math. 19 (2001), 25-53.
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