contact manifold
Let be a smooth manifold and a one form on . Then is a contact form on if
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1.
for each point , and
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2.
the restriction of the differential of is nondegenerate.
Condition 1 ensures that is a subbundle of the vector bundle . Condition 2 equivalently says is a symplectic structure on the vector bundle . A contact structure on a manifold is a subbundle of so that for each , there is a contact form defined on some neighborhood of so that . A co-oriented contact structure is a subbundle of of the form for some globally defined contact form .
A (co-oriented) contact manifold is a pair where is a manifold and is a (co-oriented) contact structure. Note, symplectic linear algebra implies that is odd. If for some positive integer , then a one form is a contact form if and only if is everywhere nonzero.
Suppose now that and are co-oriented contact manifolds. A diffeomorphism is called a contactomorphism if the pullback along of differs from by some positive smooth function , that is, .
Examples:
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1.
is a contact manifold with the contact structure induced by the one form .
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2.
Denote by the two-torus . Then, (with coordinates ) is a contact manifold with the contact structure induced by .
Title | contact manifold |
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Canonical name | ContactManifold |
Date of creation | 2013-03-22 13:43:27 |
Last modified on | 2013-03-22 13:43:27 |
Owner | RevBobo (4) |
Last modified by | RevBobo (4) |
Numerical id | 4 |
Author | RevBobo (4) |
Entry type | Definition |
Classification | msc 53D10 |
Related topic | SymplecticManifold |
Defines | contact structure |
Defines | contact form |
Defines | contactomorphism |