for each point , and
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.
is a contact manifold with the contact structure induced by the one form .
Denote by the two-torus . Then, (with coordinates ) is a contact manifold with the contact structure induced by .
|Date of creation||2013-03-22 13:43:27|
|Last modified on||2013-03-22 13:43:27|
|Last modified by||RevBobo (4)|