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Symplectization

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In mathematics, the symplectization of a contact manifold is a symplectic manifold which naturally corresponds to it.

Definition

Let ( V , ξ ) {\displaystyle (V,\xi )} be a contact manifold, and let x V {\displaystyle x\in V} . Consider the set

S x V = { β T x V { 0 } ker β = ξ x } T x V {\displaystyle S_{x}V=\{\beta \in T_{x}^{*}V-\{0\}\mid \ker \beta =\xi _{x}\}\subset T_{x}^{*}V}

of all nonzero 1-forms at x {\displaystyle x} , which have the contact plane ξ x {\displaystyle \xi _{x}} as their kernel. The union

S V = x V S x V T V {\displaystyle SV=\bigcup _{x\in V}S_{x}V\subset T^{*}V}

is a symplectic submanifold of the cotangent bundle of V {\displaystyle V} , and thus possesses a natural symplectic structure.

The projection π : S V V {\displaystyle \pi :SV\to V} supplies the symplectization with the structure of a principal bundle over V {\displaystyle V} with structure group R R { 0 } {\displaystyle \mathbb {R} ^{*}\equiv \mathbb {R} -\{0\}} .

The coorientable case

When the contact structure ξ {\displaystyle \xi } is cooriented by means of a contact form α {\displaystyle \alpha } , there is another version of symplectization, in which only forms giving the same coorientation to ξ {\displaystyle \xi } as α {\displaystyle \alpha } are considered:

S x + V = { β T x V { 0 } | β = λ α , λ > 0 } T x V , {\displaystyle S_{x}^{+}V=\{\beta \in T_{x}^{*}V-\{0\}\,|\,\beta =\lambda \alpha ,\,\lambda >0\}\subset T_{x}^{*}V,}
S + V = x V S x + V T V . {\displaystyle S^{+}V=\bigcup _{x\in V}S_{x}^{+}V\subset T^{*}V.}

Note that ξ {\displaystyle \xi } is coorientable if and only if the bundle π : S V V {\displaystyle \pi :SV\to V} is trivial. Any section of this bundle is a coorienting form for the contact structure.

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