In mathematics, the symplectization of a contact manifold is a symplectic manifold which naturally corresponds to it.
Let
(
V
,
ξ
)
be a contact manifold, and let
x
∈
V
. Consider the set
S
x
V
=
{
β
∈
T
x
∗
V
−
{
0
}
∣
ker
β
=
ξ
x
}
⊂
T
x
∗
V
of all nonzero 1forms at
x
, which have the contact plane
ξ
x
as their kernel. The union
S
V
=
⋃
x
∈
V
S
x
V
⊂
T
∗
V
is a symplectic submanifold of the cotangent bundle of
V
, and thus possesses a natural symplectic structure.
The projection
π
:
S
V
→
V
supplies the symplectization with the structure of a principal bundle over
V
with structure group
R
∗
≡
R
−
{
0
}
.
When the contact structure
ξ
is cooriented by means of a contact form
α
, there is another version of symplectization, in which only forms giving the same coorientation to
ξ
as
α
are considered:
S
x
+
V
=
{
β
∈
T
x
∗
V
−
{
0
}

β
=
λ
α
,
λ
>
0
}
⊂
T
x
∗
V
,
S
+
V
=
⋃
x
∈
V
S
x
+
V
⊂
T
∗
V
.
Note that
ξ
is coorientable if and only if the bundle
π
:
S
V
→
V
is trivial. Any section of this bundle is a coorienting form for the contact structure.