# Symplectization

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{{Unreferenced|date=December 2020}}
In [mathematics](/source/mathematics), the '''symplectization''' (or '''symplectification''') of a [contact manifold](/source/contact_manifold) is a [symplectic manifold](/source/symplectic_manifold) which naturally corresponds to it.

== Definition ==

Let <math>(V,\xi)</math> be a contact manifold, and let <math>x \in V</math>. Consider the set
: <math>S_xV = \{\beta \in T^*_xV - \{ 0 \} \mid \ker \beta = \xi_x\} \subset T^*_xV</math>
of all nonzero [1-form](/source/1-form)s at <math>x</math>, which have the contact plane <math>\xi_x</math> as their kernel. The union
:<math>SV = \bigcup_{x \in V}S_xV \subset T^*V</math>
is a [symplectic submanifold](/source/symplectic_submanifold) of the [cotangent bundle](/source/cotangent_bundle) of <math>V</math>, and thus possesses a natural symplectic structure.

The [projection](/source/projection_(mathematics)) <math>\pi : SV \to V</math> supplies the symplectization with the structure of a [principal bundle](/source/principal_bundle) over <math>V</math> with [structure group](/source/principal_bundle) <math>\R^* \equiv \R - \{0\}</math>.

== The coorientable case ==

When the [contact structure](/source/contact_structure) <math>\xi</math> is [cooriented](/source/coorientation) by means of a [contact form](/source/contact_form) <math>\alpha</math>, there is another version of symplectization, in which only forms giving the same coorientation to <math>\xi</math> as <math>\alpha</math> are considered:

:<math>S^+_xV = \{\beta \in T^*_xV - \{0\} \,|\, \beta = \lambda\alpha,\,\lambda > 0\} \subset T^*_xV,</math>

:<math>S^+V = \bigcup_{x \in V}S^+_xV \subset T^*V.</math>

Note that <math>\xi</math> is coorientable if and only if the bundle <math>\pi : SV \to V</math> is [trivial](/source/trivial_bundle). Any [section](/source/Section_(fiber_bundle)) of this bundle is a coorienting form for the contact structure.

Category:Differential topology
Category:Structures on manifolds
Category:Symplectic geometry

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Adapted from the Wikipedia article [Symplectization](https://en.wikipedia.org/wiki/Symplectization) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Symplectization?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.