Contact geometry

English

Noun

1. Template:Lb 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;
   Template:Lb the study of such structures.

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   • 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 $\alpha \wedge (d\alpha {)}^n\ne 0$. 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 $\ge$ 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.

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   • 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.

Related terms

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Translations

• German: Template:T

Further reading

• Contact manifold on The Manifold Atlas Project