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In mathematics , the symplectization of a contact manifold is a symplectic manifold which naturally corresponds to it.
Definition [ edit ] Let
(
V
,
ξ
)
{\displaystyle (V,\xi )}
x
∈
V
{\displaystyle x\in V}
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}
ξ
x
{\displaystyle \xi _{x}}
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}
The projection
π
:
S
V
→
V
{\displaystyle \pi :SV\to V}
principal bundle over
V
{\displaystyle V}
structure group
R
∗
≡
R
−
{
0
}
{\displaystyle \mathbb {R} ^{*}\equiv \mathbb {R} -\{0\}}
The coorientable case [ edit ] When the contact structure
ξ
{\displaystyle \xi }
cooriented by means of a contact form
α
{\displaystyle \alpha }
ξ
{\displaystyle \xi }
α
{\displaystyle \alpha }
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 }
π
:
S
V
→
V
{\displaystyle \pi :SV\to V}
trivial . Any section of this bundle is a coorienting form for the contact structure.
