contact geometry

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contact geometry

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contact geometry (countable and uncountable, plural contact geometries)

(differential geometry, countable) Given a smooth manifold of odd dimensionality, a distribution (subset) of the tangent bundle that satisfies the condition of complete nonintegrability, or equivalently may be locally defined as the kernel of a maximally nondegenerate differential 1-form;
(uncountable) the study of such structures.
The defining conditions for a contact geometry are opposite to two equivalent conditions for complete integrability of a hyperplane distribution: i.e. that it be tangent to a codimension 1 foliation on the manifold, whose equivalence is the content of the Frobenius theorem.

The contact geometry is in many ways an odd-dimensional counterpart of the symplectic geometry, a structure on certain even-dimensional manifolds. The concepts of contact geometry and symplectic geometry are both motivated by the mathematical formalism of classical mechanics, where one can consider either the even-dimensional phase space of a mechanical system or the constant-energy hypersurface, which, being of codimension 1, has odd dimension.
- 2004, Ko Honda, 3-Dimensional Methods in Contact Geometry, Simon Donaldson, Yakov Eliashberg, Misha Gromov (editors), Different Faces of Geometry, Springer (Kluwer Academic), page 47,
  A contact manifold $(M,\zeta )$ is a $(2n+1)$ -dimensional manifold $M$ equipped with a smooth maximally nonintegrable hyperplane field $\zeta \subset {TM}$ , i.e., locally, $\zeta =\ker \alpha$ , where $\alpha$ is a 1-form which satisfies $\alpha \wedge (d\alpha )^{n}\neq 0$ . Since $d\alpha$ is a nondegenerate 2-form when restricted to $\zeta$ , contact geometry is customarily viewed as the odd-dimensional sibling of symplectic geometry. Although contact geometry in dimensions $\geq$ 5 is still in an incipient state, contact structures in dimension 3 are much better understood, largely due to the fact that symplectic geometry in two dimensions is just the study of area.
Used other than figuratively or idiomatically: see contact,‎ geometry.
- 1998 [Kluwer Academic], L.-S. Fan, et al., Chapter 4: Sorbent Transfer and Dispersion, Barbara Toole-O'Neil (editor), Dry Scrubbing Technologies for Flue Gas Desulfurization, 1998, Springer, page 263,
  Surfaces of even the large hydrate particles have rounded protrusions and are best represented by a sphere-sphere contact geometry.
- 2003, Kurt Frischmuth, Dirk, Langemann, Distributed Numerical Calculations of Wear in the Rail-Wheel Contact, Karl Popp, Werner Schliehlen (editors), System Dynamics and Long-Term Behaviour of Railway Vehicles, Track and Subgrade, Springer, page 94,
  Along the trajectories dissipated power is calculated and projected onto the surface grid by a method using geometrical data on the contact geometry.