Contact geometry

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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.

Contents 1 Applications 2 Contact forms and structures 3 Legendrian submanifolds and knots 4 Reeb vector field 5 Some historical remarks 6 References

Applications

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'. Contact geometry has many applications to low-dimensional topology; one such connection is the fact that every three-manifold has a contact structure.

Contact forms and structures

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(e^tα)

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

Legendrian submanifolds and knots

The most interesting subspaces of a contact manifold are its Legendrian submanifolds. The non-integrability of the contact hyperplane field on a (2n+1)-dimensional manifold means that no 2n-dimensional submanifold has it as its tangent bundle, not even locally. However, it is in general possible to find n-dimensional (embedded or immersed) submanifolds whose tangent spaces lie inside the contact field. Legendrian submanifolds are analogous to Lagrangian submanifolds of symplectic manifolds. There is a precise relation: the lift of a Legendrian submanifold in a symplectization of a contact manifold is a Lagrangian submanifold. The simplest example of Legendrian submanifolds are Legendrian knots inside a contact three-manifold. Inequivalent Legendrian knots may be equivalent as smooth knots.

Legendrian submanifolds are very rigid objects; in some situations, being Legendrian forces submanifolds to be unknotted. Symplectic field theory provides invariants of Legendrian submanifolds called relative contact homology that can sometimes distinguish distinct Legendrian submanifolds that are topologically identical.

Reeb vector field

If α is a contact form for a given contact structure, the Reeb vector field R can be defined as the unique element of the kernel of dα such that α(R)=1. Its dynamics can be used to study the structure of the contact manifold or even the underlying manifold using techniques of Floer homology such as symplectic field theory and embedded contact homology.

Some historical remarks

The roots of contact geometry appear in work of Christiaan Huygens, Barrow and Isaac Newton. The theory of contact transformations (i.e. transformations preserving a contact structure) was developed by Sophus Lie, with the dual aims of studying differential equations (e.g. the Legendre transformation) and describing the 'change of space element', familiar from projective duality.

References

Introductions to contact geometry:

• Etnyre, J. Introductory lectures on contact geometry, Proc. Sympos. Pure Math. 71 (2003), 81-107.arXiv
• Geiges, H. Contact Geometry, arXiv
• Aebischer et.al. symplectic geometry, Birkhäuser, 1994.

Contact three-manifolds and Legendrian knots:

• William Thurston, Three-Dimensional Geometry and Topology. Princeton University Press, 1997.

Information on the history of contact geometry:

• 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.
• Arnold, V.I. (trans. E. Primrose), Huygens and Barrow, Newton and Hooke: pioneers in mathematical analysis and catastrophe theory from evolvents to quasicrystals. Birkhauser Verlag, 1990.ru:Контактная структура

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