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Construction junctions are diamonds. They show where multiple structures must coexist on the same carrier and satisfy compatibility conditions.

Canonical static concept record

Contact manifold

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Summary

An odd-dimensional manifold with a maximally nonintegrable hyperplane distribution.

Record metadata

Canonical name: Contact manifold
Concept ID: contact_manifold
Content version: 2.3.0
Primary field: Mathematics
Fields: Mathematics
Primary domain: Differential geometry
Domains: Differential geometry, Topology
Alternate terminology: No alternate terminology is currently authored for this record.

Carrier(s)

smooth manifold $M$

Data

smooth hyperplane distribution $\xi\subset TM$

Axioms / constraints

locally $\xi=\ker\alpha$ with $d\alpha|_\xi$ nondegenerate (equivalently $\alpha\wedge(d\alpha)^n$ nowhere zero)

Notes

A globally defined contact form is extra coorientation/trivialization data. The contact distribution is the intrinsic object.

Concept sources

Wikipedia — Contact geometry — Contact geometry · expository · source ID wp-contact-manifold
nLab — Contact manifold — Contact manifold · reference · source ID nlab-contact-manifold
DOI — The regularity and products in contact geometry — The regularity and products in contact geometry · research article · source ID grabowska-grabowski-contact

Incoming relations (arrows to this concept)

Each relation below ends at this concept.

Smooth manifold Contact manifold

Permalink to relation

This is an authored directed relation from the source endpoint to the target endpoint.

Relation ID: e_smooth_contact
Relation type: Add data add-data
Direction: source → target
Endpoint roles: source: Builds toward; target: Built from
Authored annotation: + contact distribution

Authored explanation

Add a maximally nonintegrable hyperplane field.

How to interpret this relation type

Equip an existing carrier or structured object with additional chosen data, when such compatible data exists. Use a construction junction when several independently meaningful inputs must coexist on the same carrier or interact compatibly.

Relation sources

Wikipedia — Contact geometry — Contact geometry · expository · source ID wp-contact-manifold
nLab — Contact manifold — Contact manifold · reference · source ID nlab-contact-manifold
Lee — Introduction to Smooth Manifolds — Introduction to Smooth Manifolds · textbook · source ID lee-smooth-manifolds

Outgoing relations (arrows from this concept)

Each relation below starts at this concept.

No direct relations are authored in this direction.

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