contact geometry

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contact geometry

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contact geometry (countable and uncountable, plural contact geometries)

1. (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 ${\displaystyle (M,\zeta )}$ is a ${\displaystyle (2n+1)}$-dimensional manifold ${\displaystyle M}$ equipped with a smooth maximally nonintegrable hyperplane field ${\displaystyle \zeta \subset {TM}}$, i.e., locally, ${\displaystyle \zeta =\ker \alpha }$, where ${\displaystyle \alpha }$ is a 1-form which satisfies ${\displaystyle \alpha \wedge (d\alpha )^{n}\neq 0}$. Since ${\displaystyle d\alpha }$ is a nondegenerate 2-form when restricted to ${\displaystyle \zeta }$, contact geometry is customarily viewed as the odd-dimensional sibling of symplectic geometry. Although contact geometry in dimensions ${\displaystyle \geq }$ 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.

2. Used other than figuratively or idiomatically: see contact,‎ geometry.
   • 1998 , 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.

• contact manifold

• Contact manifold on The Manifold Atlas Project