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In mathematics , contact geometry is the study of a geometric structure on smooth manifolds given by a hyperplane distribution in the tangent bundle satisfying a condition called 'complete non-integrability'. Equivalently, such a distribution may be given at least locally as the kernel of a differential one-form, and the non-integrability condition translates into a maximal non-degeneracy condition on the form. These conditions are opposite to two equivalent conditions for ' complete integrability ' of a hyperplane distribution, i. Contact geometry is in many ways an odd-dimensional counterpart of symplectic geometry , a structure on certain even-dimensional manifolds.

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 constant-energy hypersurface, which, being codimension one, has odd dimension.

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Like symplectic geometry, contact geometry has broad applications in physics , e. 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 , by Michael Hutchings to define an invariant of smooth three-manifolds, and by Lenhard Ng to define invariants of knots.

It was also used by Yakov Eliashberg to derive a topological characterization of Stein manifolds of dimension at least six. A contact structure on an odd dimensional manifold is a smoothly varying family of codimension one subspaces of each tangent space of the manifold, satisfying a non-integrability condition. The family may be described as a section of a bundle as follows:. The non-integrability condition can be given explicitly as: [1].

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.

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This construction provides any contact manifold M with a natural symplectic bundle of rank 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.

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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. Choose a Riemannian metric on the manifold N and let H be the associated kinetic energy. Then the Liouville form restricted to the unit cotangent bundle is a contact structure. This corresponds to a special case of the second construction, where the flow of the Euler vector field Y corresponds to linear scaling of momenta p's, leaving the q's fixed.

The vector field R , defined by the equalities. More precisely, using the Riemannian metric, one can identify each point of the cotangent bundle of N with a point of the tangent bundle of N , and then the value of R at that point of the unit cotangent bundle is the corresponding unit vector parallel to N. This new manifold is called the symplectization sometimes symplectification in the literature of the contact manifold M.

Conformal and related changes of metric on the product of two almost contact metric manifolds. Paris , , 34 , — Geometry of Submanifolds.

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Marcel Dekker , New York , Geometry of warped products as Riemannian submanifolds and related problems. Curvature relations in trans-Sasakian manifolds. II Portuguese Braga , Minho , Braga , , — Invariant submanifolds of a trans-Sasakian manifold. Debrecen , , 38 , — On three-dimensional trans-Sasakian manifolds. Extracta Mathematicae , , 23 , — On pseudo-slant submanifolds of trans-Sasakian manifolds. Estonian Acad. Almost contact manifolds with torsion and parallel spinors.

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Riemannian Geometry of Contact and Symplectic Manifolds - David E. Blair - Google книги

Invariant submanifolds and modes of non-linear autonomous systems. The sixteen classes of almost Hermitian manifolds and their linear invariants. Pura Appl. Almost contact structures and curvature tensors.

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Kodai Math. Semi-invariant submanifolds of a certain class of almost contact metric manifolds.

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Tensor N. Invariant submanifolds of normal contact metric manifolds.