1Symplectic manifolds
III Symplectic Geometry
1.1 Symplectic linear algebra
In symplectic geometry, we study symplectic manifolds. These are manifolds
equipped with a certain structure on the tangent bundle. In this section, we first
analyze the condition fiberwise.
Definition
(Symplectic vector space)
.
A symplectic vector space is a real vector
space
V
together with a non-degenerate skew-symmetric bilinear map Ω :
V ×
V → R.
Recall that
Definition
(Non-degenerate bilinear map)
.
We say a bilinear map Ω is non-
degenerate if the induced map
˜
Ω : V → V
∗
given by v 7→ Ω(v, ·) is bijective.
Even if we drop the non-degeneracy condition, there aren’t that many sym-
plectic vector spaces around up to isomorphism.
Theorem
(Standard form theorem)
.
Let
V
be a real vector space and Ω a skew-
symmetric bilinear form. Then there is a basis
{u
1
, . . . , u
k
, e
1
, . . . , e
n
, f
1
, . . . , f
n
}
of V such that
(i) Ω(u
i
, v) = 0 for all v ∈ V
(ii) Ω(e
i
, e
j
) = Ω(f
i
, f
j
) = 0.
(iii) Ω(e
i
, f
j
) = δ
ij
.
The proof is a skew-symmetric version of Gram–Schmidt.
Proof. Let
U = {u ∈ V : Ω(u, v) = 0 for all v ∈ V },
and pick a basis u
1
, . . . , u
k
of this. Choose any W complementary to U.
First pick
e
1
∈ W \ {
0
}
arbitrarily. Since
e
1
6∈ U
, we can pick
f
1
such that
Ω(e
1
, f
1
) = 1. Then define W
1
= span{e
i
, f
i
}, and
W
Ω
1
= {w ∈ W : Ω(w, v) = 0 for all v ∈ W
1
}.
It is clear that
W
1
∩ W
Ω
1
=
{
0
}
. Moreover,
W
=
W
1
⊕ W
Ω
1
. Indeed, if
v ∈ W
,
then
v = (Ω(v, f
1
)e
1
− Ω(v, e
1
)f
1
) + (v −(Ω(v, f
1
)e
1
− Ω(v, e
1
)f
1
)),
Then we are done by induction on the dimension.
Here
k
=
dim U
and 2
n
=
dim V − k
are invariants of (
V,
Ω). The number
2n is called the rank of Ω. Non-degeneracy is equivalent to k = 0.
Exercise. Ω is non-degenerate iff Ω
∧n
= Ω ∧ ··· ∧ Ω 6= 0.
By definition, every symplectic vector space is canonically isomorphic to
its dual, given by the map
˜
Ω
. As in the above theorem, a symplectic basis
{e
1
, . . . , e
n
, f
1
, . . . , f
n
} of V is a basis such that
Ω(e
i
, e
j
) = Ω(f
i
, f
j
) = 0, Ω(e
i
, f
i
) = δ
ij
.
Every symplectic vector space has a symplectic basis, and in particular has even
dimension. In such a basis, the matrix representing Ω is
Ω
0
=
0 I
−I 0
.
We will need the following definitions:
Definition
(Symplectic subspace)
.
If (
V,
Ω) is a symplectic vector space, a
symplectic subspace is a subspace W ⊆ V such that Ω|
W ×W
is non-degenerate.
Definition
(Isotropic subspace)
.
If (
V,
Ω) is a symplectic vector space, an
isotropic subspace is a subspace W ⊆ V such that Ω|
W ×W
= 0.
Definition
(Lagrangian subspace)
.
If (
V,
Ω) is a symplectic vector space, an
Lagrangian subspace is an isotropic subspace W with dim W =
1
2
dim V .
Definition
(Symplectomorphism)
.
A symplectomorphism between symplectic
vector spaces (
V,
Ω)
,
(
V
0
,
Ω
0
) is an isomorphism
ϕ
:
V → V
0
such that
ϕ
∗
Ω
0
= Ω.
The standard form theorem says any two 2
n
-dimensional symplectic vector
space (V, Ω) are symplectomorphic to each other.