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| In [[mathematics]], the '''symplectization''' of a [[contact manifold]] is a [[symplectic manifold]] which naturally corresponds to it.
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| == Definition ==
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| Let <math>(V,\xi)</math> be a contact manifold, and let <math>x \in V</math>. Consider the set
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| : <math>S_xV = \{\beta \in T^*_xV - \{ 0 \} \mid \ker \beta = \xi_x\} \subset T^*_xV</math>
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| of all nonzero [[1-form]]s at <math>x</math>, which have the contact plane <math>\xi_x</math> as their kernel. The union
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| :<math>SV = \bigcup_{x \in V}S_xV \subset T^*V</math>
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| is a [[symplectic submanifold]] of the [[cotangent bundle]] of <math>V</math>, and thus possesses a natural symplectic structure.
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| The [[projection (mathematics)|projection]] <math>\pi : SV \to V</math> supplies the symplectization with the structure of a [[principal bundle]] over <math>V</math> with [[principal bundle|structure group]] <math>\R^* \equiv \R - \{0\}</math>.
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| == The coorientable case ==
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| When the [[contact structure]] <math>\xi</math> is [[coorientation|cooriented]] by means of a [[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:
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| :<math>S^+_xV = \{\beta \in T^*_xV - \{0\} \,|\, \beta = \lambda\alpha,\,\lambda > 0\} \subset T^*_xV,</math> | |
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| :<math>S^+V = \bigcup_{x \in V}S^+_xV \subset T^*V.</math>
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| Note that <math>\xi</math> is coorientable if and only if the bundle <math>\pi : SV \to V</math> is [[trivial bundle|trivial]]. Any [[Section (fiber bundle)|section]] of this bundle is a coorienting form for the contact structure.
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| [[Category:Differential topology]]
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| [[Category:Structures on manifolds]]
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| [[Category:Symplectic geometry]]
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