In mathematics, contact geometry is the study of a geometric structure on smooth manifolds given by a hyperplane distribution in the tangent bundle and specified by a one-form, both of which satisfy a 'maximum non-degeneracy' condition called 'complete non-integrability'. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and A differentiable manifold is a type of Manifold that is locally similar enough to Euclidean space to allow one to do Calculus. In Differential geometry, a discipline within Mathematics, a distribution is a subset of the Tangent bundle of a manifold satisfying certain In Mathematics, the tangent bundle of a smooth (or differentiable manifold M, denoted by T ( M) or just TM, is the In the mathematical fields of Differential geometry and Tensor calculus, differential forms are an approach to Multivariable calculus which is From the Frobenius theorem, one recognizes the condition as the opposite of the condition that the distribution be determined by a codimension one foliation on the manifold ('complete integrability'). In Mathematics, Frobenius' theorem gives Necessary and sufficient conditions for finding a maximal set of independent solutions of an Overdetermined system In Mathematics, a foliation is a geometric device used to study manifolds Informally speaking a foliation is a kind of "clothing" worn on a manifold

Contact geometry is in many ways an odd-dimensional counterpart of symplectic geometry, which belongs to the even-dimensional world. Symplectic geometry is a branch of differential topology/geometry which studies Symplectic manifolds that is Differentiable manifolds equipped with a Both contact and symplectic geometry are motivated by the mathematical formalism of classical mechanics, where one can consider either the even-dimensional phase space of a mechanical system or the odd-dimensional extended phase space that includes the time variable. Classical mechanics is used for describing the motion of Macroscopic objects from Projectiles to parts of Machinery, as well as Astronomical objects In Mathematics and Physics, a phase space, introduced by Willard Gibbs in 1901 is a Space in which all possible states of a System

Contents 1 Applications 2 Contact forms and structures 2.1 Properties 2.2 Relation with symplectic structures 2.3 Examples 3 Legendrian submanifolds and knots 4 Reeb vector field 5 Some historical remarks 6 References 6.1 Introductions to contact geometry 6.2 Applications to differential equations 6.3 Contact three-manifolds and Legendrian knots 6.4 Information on the history of contact geometry

Applications

Contact geometry has — as does symplectic geometry — broad applications in physics, e. Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. g. geometrical optics, classical mechanics, thermodynamics, geometric quantization, and applied mathematics such as control theory. Classical mechanics is used for describing the motion of Macroscopic objects from Projectiles to parts of Machinery, as well as Astronomical objects In Physics, thermodynamics (from the Greek θερμη therme meaning " Heat " and δυναμις dynamis meaning " In Mathematical physics, geometric quantization is a mathematical approach to defining a quantum theory corresponding to a given Classical theory. Control theory is an interdisciplinary branch of Engineering and Mathematics, that deals with the behavior of Dynamical systems The desired output One can prove amusing things, like 'You can always parallel-park your car, provided the space is big enough'. Parallel parking is a method of Parking a Vehicle in line with other parked cars Contact geometry also has applications to low-dimensional topology; for example, it has been used by Kronheimer and Mrowka to prove the property P conjecture and by Gompf to derive a topological characterization of Stein manifolds. In Mathematics, low-dimensional topology is the branch of Topology that studies Manifolds of four or fewer dimensions Peter Benedict Kronheimer is a British Mathematician, known for his work on Gauge theory and its applications to 3- and 4-dimensional topology In Mathematics, the Property P conjecture is a statement about 3-manifolds obtained by Dehn surgery on a knot in the 3-sphere. In Mathematics, a Stein manifold in the theory of Several complex variables and Complex manifolds is a complex Submanifold of the Vector

Contact forms and structures

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

A contact structure on a manifold is hyperplane field (or distribution of hyperplanes) ξ in the tangent bundle which is the kernel of a contact form α. A differentiable manifold is a type of Manifold that is locally similar enough to Euclidean space to allow one to do Calculus. In the mathematical fields of Differential geometry and Tensor calculus, differential forms are an approach to Multivariable calculus which is In Mathematics, the tangent bundle of a smooth (or differentiable manifold M, denoted by T ( M) or just TM, is the The contact form α is defined for a given contact structure only up to multiplication by a nowhere vanishing smooth function.

Properties

It follows from the Frobenius theorem on integrability that the contact field ξ is completely nonintegrable. In Mathematics, Frobenius' theorem gives Necessary and sufficient conditions for finding a maximal set of independent solutions of an Overdetermined system This property of the contact field is roughly the opposite of being a field formed by the tangent planes to a family of nonoverlapping hypersurfaces in M. In particular, you cannot find a piece of a hypersurface tangent to ξ on an open set of M. More precisely, a maximally integrable subbundle has dimension n.

Relation with symplectic structures

A consequence of the definition is that the restriction of the 2-form ω = d α to a hyperplane in ξ is a nondegenerate 2-form. This construction provides any contact manifold M with a natural symplectic bundle of dimension one smaller than the dimension of M. Note that a symplectic vector space is always even-dimensional, while contact manifolds need to be odd-dimensional.

The cotangent bundle T^*N of any n-dimensional manifold N is itself a manifold (of dimension 2n) and supports naturally an exact symplectic structure ω = d λ. In Mathematics, especially Differential geometry, the cotangent bundle of a Smooth manifold is the Vector bundle of all the Cotangent spaces (This 1-form λ is sometimes called the Liouville form). In Mathematics, the tautological one-form is a special 1-form defined on the Cotangent bundle T * Q of a Manifold Q There are two main ways to construct an associated contact manifold, one of dimension 2n - 1, one of dimension 2n + 1.

1. Let M be the projectivization of the cotangent bundle of N: thus M is fiber bundle over a M whose fiber at a point x is the space of lines in T^*N, or, equivalently, the space of hyperplanes in TN. In Mathematics a projective space is a set of elements constructed from a vector space such that a distinct element of the projective space consists of all non-zero vectors which The 1-form λ does not descend to a genuine 1-form on M. However, it is homogeneous of degree 1, and so it defines a 1-form with values in the line bundle O(1), which is the dual of the fibrewise tautological line bundle of M. The kernel of this 1-form defines a contact distribution. There are several ways of making this construction more explicit by making choices.
   • Suppose that H is a smooth function on T^*N. Then each level set,

   (q,p) ∈T^*N : H(q,p) = E,

   is a contact manifold of dimension 2n-1 at its smooth points (i. e. the points where the differential of H does not vanish). The contact form α is the restriction of the Liouville form to the level set. This construction originates in Hamiltonian mechanics, where H is a Hamiltonian of a mechanical system with the configuration space N and the phase space T^*N, and E is the value of the energy. Hamiltonian mechanics is a re-formulation of Classical mechanics that was introduced in 1833 by Irish mathematician William Rowan Hamilton.

   • Choose a Riemannian metric on the manifold N. In Riemannian geometry, a Riemannian manifold ( M, g) (with Riemannian metric g) is a real Differentiable manifold M That allows one to consider the unit sphere in each cotangent plane, resulting in a unit cotangent bundle of N, which is a smooth manifold of dimension 2n-1. In Mathematics, a unit Sphere is the set of points of Distance 1 from a fixed central point where a generalized concept of distance may be used a closed Then the Liouville form restricted to the unit cotangent bundle is a contact structure. This corresponds to a special case of the first construction, where the Hamiltonian is taken to be the square of the length of a vector in the cotangent bunle. The vector field R, defined by the equalities

   λ(R) = 1 and d λ(R, A) = 0 for all vector fields A,

   is called the Reeb vector field, and it generates the geodesic flow of the Riemannian metric. In Mathematics a vector field is a construction in Vector calculus which associates a vector to every point in a (locally Euclidean space. In Mathematics, a geodesic /ˌdʒiəˈdɛsɪk -ˈdisɪk/ -dee-sik is a generalization of the notion of a " straight line " to " curved spaces

2. On the other hand, one can build a contact manifold M of dimension 2n+1 by considering the first jet bundle of the real valued functions on N. In Differential geometry, the jet bundle is a certain construction which makes a new smooth Fiber bundle out of a given smooth fiber bundle This bundle is isomorphic to T^*N×R using the exterior derivative of a function. In Differential geometry, the exterior derivative extends the concept of the differential of a function which is a form of degree zero to Differential forms With coordinates (x, t), M has a contact structure

   α = dt + λ.

Conversely, given any contact manifold M, the product M×R has a natural structure of a symplectic manifold. If α is a contact form on M, then

ω = d(e^tα)

is a symplectic form on M×R, where t denotes the variable in the R-direction. This new manifold is called the symplectization (sometimes symplectification in the literature) of the contact manifold M. In Mathematics, the symplectization of a Contact manifold is a Symplectic manifold which naturally corresponds to it

Examples

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

(x, y, z),

the 1-form

dz − y dx.

The contact plane ξ at a point

(x, y, z)

is spanned by the vectors

X₁ = ∂_y

and

X₂ = ∂_x + y ∂_z.

By replacing the single variables x and y with the multivariables x₁,. . . ,x_n, y₁,. . . ,y_n, one can generalize this example to any R²ⁿ⁺¹. By a theorem of Darboux, every contact structure on a manifold looks locally like this particular contact structure on the 2n+1-dimensional vector space. Darboux's theorem is a Theorem in the mathematical field of Differential geometry and more specifically Differential forms, partially generalizing

An important class of contact manifolds is formed by Sasakian manifolds. In Differential geometry, a Sasakian manifold is a Contact manifold (M\theta equipped with a special kind of Riemannian metric g

Legendrian submanifolds and knots

The most interesting subspaces of a contact manifold are its Legendrian submanifolds. In Mathematics, contact geometry is the study of a geometric structure on Smooth manifolds given by a hyperplane distribution in the Tangent bundle 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. Floer homology is a mathematical tool used in the study of Symplectic geometry and low-dimensional Topology. In Mathematics, in the area of Symplectic topology, relative contact homology is an invariant of spaces together with a chosen subspace

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. Floer homology is a mathematical tool used in the study of Symplectic geometry and low-dimensional Topology. Floer homology is a mathematical tool used in the study of Symplectic geometry and low-dimensional Topology.

Some historical remarks

The roots of contact geometry appear in work of Christiaan Huygens, Barrow and Isaac Newton. Christiaan Huygens (ˈhaɪgənz in English ˈhœyɣəns in Dutch) ( April 14, 1629 &ndash July 8, 1695) was a Dutch Sir Isaac Newton, FRS (ˈnjuːtən 4 January 1643 31 March 1727) Biography Early years See also Isaac Newton's early life and achievements 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. Marius Sophus Lie (liː as "Lee" ( 17 December 1842 - 18 February 1899) was a Norwegian -born Mathematician. g. the Legendre transformation or canonical transformation) and describing the 'change of space element', familiar from projective duality. In Mathematics, it is often desirable to express a functional relationship f(x\ as a different function whose argument is the derivative of f   rather In Hamiltonian mechanics, a canonical transformation is a change of Canonical coordinates (\mathbf{q} \mathbf{p} t \rightarrow (\mathbf{Q} \mathbf{P} t In the Geometry of the Projective plane, duality refers to geometric transformations that replace points by lines and lines by points while preserving

References

Introductions to contact geometry

• Etnyre, J. Introductory lectures on contact geometry, Proc. Sympos. Pure Math. 71 (2003), 81–107, math. SG/0111118
• Geiges, H. Contact Geometry, math. SG/0307242
• Aebischer et. al. Symplectic geometry, Birkhäuser (1994), ISBN 3764350644
• V. I. Arnold, Mathematical Methods of Classical Mechanics, Springer-Verlag (1989), ISBN 0-387-96890-3

Applications to differential equations

• V. I. Arnold, Geometrical Methods In The Theory Of Ordinary Differential Equations, Springer-Verlag (1988), ISBN 0-387-96649-8

Contact three-manifolds and Legendrian knots

• William Thurston, Three-Dimensional Geometry and Topology. Princeton University Press(1997), ISBN 0691083045

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.

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