Symplectic manifolds
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− | == Introduction == | + | |
<wikitex>; | <wikitex>; | ||
A '''symplectic manifold''' is a smooth manifold $M$ together with a differential two-form $\omega$ that is nondegenerate and closed. The form $\omega$ is called a '''symplectic form'''. The nondegeneracy means that the highest nonzero power of $\omega$ is a volume form on $M.$ It follows that a symplectic manifold is even dimensional. | A '''symplectic manifold''' is a smooth manifold $M$ together with a differential two-form $\omega$ that is nondegenerate and closed. The form $\omega$ is called a '''symplectic form'''. The nondegeneracy means that the highest nonzero power of $\omega$ is a volume form on $M.$ It follows that a symplectic manifold is even dimensional. | ||
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Symplectic manifolds originated from classical mechanics. The phase space of a dynamical system is the cotangent bundle of the configuration space and it is equipped with a symplectic form. This symplectic form is preserved by the flow of the system. | Symplectic manifolds originated from classical mechanics. The phase space of a dynamical system is the cotangent bundle of the configuration space and it is equipped with a symplectic form. This symplectic form is preserved by the flow of the system. | ||
</wikitex> | </wikitex> | ||
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== Examples == | == Examples == | ||
− | <wikitex> | + | <wikitex>; |
$\bullet$ | $\bullet$ | ||
The most basic example of a symplectic manifold is $\mathbb R^{2n}$ equipped with the form | The most basic example of a symplectic manifold is $\mathbb R^{2n}$ equipped with the form | ||
$\omega_0:=dx^1\wedge dy^1 + \ldots + dx^n\wedge dy^n.$ | $\omega_0:=dx^1\wedge dy^1 + \ldots + dx^n\wedge dy^n.$ | ||
− | A theorem of Darboux {{cite|McDuff | + | A theorem of Darboux {{cite|McDuff&Salamon1998}} states that locally every |
symplectic manifold if of this form. More precisely, if $(M,\omega)$ is a symplectic $2n$-manifold | symplectic manifold if of this form. More precisely, if $(M,\omega)$ is a symplectic $2n$-manifold | ||
then for every point $x\in M$ there exists an open neighbourhood $U\subset M$ of $p$ and a | then for every point $x\in M$ there exists an open neighbourhood $U\subset M$ of $p$ and a | ||
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$\bullet$ | $\bullet$ | ||
− | The complex projective space $\mathbb C \mathbb P^n$ is symplectic with respect to its | + | The complex projective space $\mathbb C \mathbb P^n$ is symplectic with respect to its Kähler form. |
Its pull back to a complex projective smooth manifold $X \subset \mathbb C \mathbb P^n$ is also symplectic. | Its pull back to a complex projective smooth manifold $X \subset \mathbb C \mathbb P^n$ is also symplectic. | ||
− | More generally, every | + | More generally, every Kähler manifold is symplectic. |
</wikitex> | </wikitex> | ||
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== Symmetries == | == Symmetries == | ||
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a group $\operatorname{Ham}(M,\omega)$ called the group of Hamiltonian diffeomorphism. Its Lie algebra can be | a group $\operatorname{Ham}(M,\omega)$ called the group of Hamiltonian diffeomorphism. Its Lie algebra can be | ||
identified with the quotient of the space of smooth functions on $M$ by the constants. | identified with the quotient of the space of smooth functions on $M$ by the constants. | ||
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</wikitex> | </wikitex> | ||
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==Constructions== | ==Constructions== | ||
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===Products=== | ===Products=== | ||
+ | <wikitex>; | ||
The product of symplectic manifolds $(M_1,\omega_1)$ and $(M_2,\omega_2)$ is a symplectic manifold with | The product of symplectic manifolds $(M_1,\omega_1)$ and $(M_2,\omega_2)$ is a symplectic manifold with | ||
respect to the form $a\cdot p_1^*\omega_1 + b\cdot p_2^*\omega_2$ for nonzero real numbers | respect to the form $a\cdot p_1^*\omega_1 + b\cdot p_2^*\omega_2$ for nonzero real numbers | ||
$a,b\in \mathbb R.$ Here $p_i\colon M_1\times M_2\to M_i$ is the projection. | $a,b\in \mathbb R.$ Here $p_i\colon M_1\times M_2\to M_i$ is the projection. | ||
+ | </wikitex> | ||
===Bundles=== | ===Bundles=== | ||
+ | <wikitex>; | ||
A locally trivial bundle $M\to E\to B$ is called symplectic (resp. Hamiltonian) if its structure | A locally trivial bundle $M\to E\to B$ is called symplectic (resp. Hamiltonian) if its structure | ||
group is a subgroup of the group of symplectic (resp. Hamiltonian) diffeomorphisms. | group is a subgroup of the group of symplectic (resp. Hamiltonian) diffeomorphisms. | ||
− | |||
'''Example.''' The product of the Hopf bundle with the circle is a symplectic bundle $T^2 \to S^3 \times S^1 \to S^2.$ | '''Example.''' The product of the Hopf bundle with the circle is a symplectic bundle $T^2 \to S^3 \times S^1 \to S^2.$ | ||
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admits a fiberwise symplectic form on the total space. | admits a fiberwise symplectic form on the total space. | ||
{{endthm}} | {{endthm}} | ||
− | + | Using this theorem, one can construct examples of symplectic fiber bundles with fiberwise symplectic form on the total space (see examples below). | |
'''Example (twistor bundles)''' | '''Example (twistor bundles)''' | ||
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is a symplectic fiber bundle with fiberwise symplectic structure. This construction can be compactified by taking lattices in $G$ which intersect trivially with $K$. A particular example is given by the fiber bundle | is a symplectic fiber bundle with fiberwise symplectic structure. This construction can be compactified by taking lattices in $G$ which intersect trivially with $K$. A particular example is given by the fiber bundle | ||
$$SO(2n)/U(n)\rightarrow SO(2n,p)/U(n)\times SO(p)\rightarrow SO(2n,p)/SO(2n)\times SO(p),$$ | $$SO(2n)/U(n)\rightarrow SO(2n,p)/U(n)\times SO(p)\rightarrow SO(2n,p)/SO(2n)\times SO(p),$$ | ||
− | and its compactification by lattices. | + | and its compactification by lattices.</wikitex> |
===Symplectic reduction=== | ===Symplectic reduction=== | ||
+ | <wikitex>; | ||
Let $G$ be a Lie group acting on a symplectic manifold $(M,\omega)$ in a hamiltonian way. Denote | Let $G$ be a Lie group acting on a symplectic manifold $(M,\omega)$ in a hamiltonian way. Denote | ||
by $\mu: M\rightarrow\frak{g}^*$ the moment map of this action. Since $G$ acts on the level set | by $\mu: M\rightarrow\frak{g}^*$ the moment map of this action. Since $G$ acts on the level set | ||
− | $\mu^{-1}(a),a\in\frak{g}^*$, one can consider the orbit space $\mu^{-1}(a)/G$. It is an orbifold in general, but it happens to be a manifold, when $G$ acts freely on the preimage, and $a$ is a regular point. In this case, $\tilde M=\mu^{-1}(a)/G$ is a symplectic manifold as well, called '''symplectic reduction'''. It is often denoted by $M//G$ | + | $\mu^{-1}(a),a\in\frak{g}^*$, one can consider the orbit space $\mu^{-1}(a)/G$. It is an orbifold in general, but it happens to be a manifold, when $G$ acts freely on the preimage, and $a$ is a regular point. In this case, $\tilde M=\mu^{-1}(a)/G$ is a symplectic manifold as well, called '''symplectic reduction'''. It is often denoted by $M//G$. |
− | + | </wikitex> | |
===Symplectic cut=== | ===Symplectic cut=== | ||
− | Let $(M,\omega)$ be a symplectic manifold with a hamiltonian action of the circle $S^1.$ If $\mu:\rightarrow \mathbb R$ is the moment map, $M_a=\mu^{-1}(a)$ is a regular level, then the action restricted to $M_a$ has no fixed points, hence $M_a$ is the boundary of the associated disk bundle W. This is a manifold if the action is free and an orbifold if a non-trivial isotropy occurs. | + | <wikitex>; |
− | + | Let $(M,\omega)$ be a symplectic manifold with a hamiltonian action of the circle $S^1.$ If $\mu:\rightarrow \mathbb R$ is the moment map, $M_a=\mu^{-1}(a)$ is a regular level, then the action restricted to $M_a$ has no fixed points, hence $M_a$ is the boundary of the associated disk bundle W. This is a manifold if the action is free and an orbifold if a non-trivial isotropy occurs. | |
+ | </wikitex> | ||
===Coadjoint orbits=== | ===Coadjoint orbits=== | ||
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===Donaldson's theorem on submanifolds=== | ===Donaldson's theorem on submanifolds=== | ||
− | ===Surgery=== | + | ===Surgery in codimension 2=== |
+ | <wikitex>; | ||
Consider two symplectic manifolds $(M_1,\omega_1),(M_2,\omega_2)$ of equal dimension and suppose that there are codimension two symplectic submanifolds $V_1\subset M_1,V_2\subset M_2$ and a symplectomorphism $f:V_1\rightarrow V_2$ such that Chern classes of normal bundles satisfy $f^*c_1(\nu_2)=-c_1(\nu_1).$ | Consider two symplectic manifolds $(M_1,\omega_1),(M_2,\omega_2)$ of equal dimension and suppose that there are codimension two symplectic submanifolds $V_1\subset M_1,V_2\subset M_2$ and a symplectomorphism $f:V_1\rightarrow V_2$ such that Chern classes of normal bundles satisfy $f^*c_1(\nu_2)=-c_1(\nu_1).$ | ||
Then by removing tubular neighborhods of $V_1$ and $V_2$ we get manifolds with boundaries. The map $f$ induces a diffeomorphism of the boundaries, one can form a new manifold identifying the boundaries by this diffeomorphism and define on it a symplectic form which coincides with | Then by removing tubular neighborhods of $V_1$ and $V_2$ we get manifolds with boundaries. The map $f$ induces a diffeomorphism of the boundaries, one can form a new manifold identifying the boundaries by this diffeomorphism and define on it a symplectic form which coincides with | ||
− | $\omega_1$ and $\omega_2$ outside of a tubular neigborhood of the trace of glueing {{cite|Gompf1995}}. The same works if $V_1,V_2$ are symplectic submanifolds of a connected symplectic manifold. | + | $\omega_1$ and $\omega_2$ outside of a tubular neigborhood of the trace of glueing {{cite|Gompf1995}}. The same works if $V_1,V_2$ are symplectic submanifolds of a connected symplectic manifold. </wikitex> |
− | + | ===Symplectic blow-up=== | |
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== Invariants == | == Invariants == |
Latest revision as of 11:47, 16 May 2013
This page has not been refereed. The information given here might be incomplete or provisional. |
Contents |
1 Introduction
A symplectic manifold is a smooth manifold together with a differential two-form
that is nondegenerate and closed. The form
is called a symplectic form. The nondegeneracy means that the highest nonzero power of
is a volume form on
It follows that a symplectic manifold is even dimensional.
Symplectic manifolds originated from classical mechanics. The phase space of a dynamical system is the cotangent bundle of the configuration space and it is equipped with a symplectic form. This symplectic form is preserved by the flow of the system.
2 Examples
The most basic example of a symplectic manifold is
equipped with the form
A theorem of Darboux [McDuff&Salamon1998] states that locally every
symplectic manifold if of this form. More precisely, if is a symplectic
-manifold
then for every point
there exists an open neighbourhood
of
and a
diffeomorphism
such that the restriction of
to
is equal to the pull-back
This implies that symplectic manifolds have
no local invariants.
An area form on an oriented surface is symplectic.
Let
be a smooth manifold and let
be a one-form on the cotangent bundle
defined as follows.
If
is a vector tangent to
at a point
then
where
is the projection. In local coordinates the form
can be expressed as
The differential
is a symplectic form on the cotangent bundle
If
is a closed, i.e. compact and without boundary, symplectic
-manifold then the cohomology classes
are non-zero for
This follows from the fact that the cohomology class of the volume
form
is nonzero on a closed manifold. This necessary condition implies that spheres of dimension greater than
two are not symplectic. More generally, no closed manifold of the form
is symplectic for
The complex projective space
is symplectic with respect to its Kähler form.
Its pull back to a complex projective smooth manifold
is also symplectic.
More generally, every Kähler manifold is symplectic.
3 Symmetries
A diffeomorphism of a symplectic manifold
is called symplectic if it preserves
the symplectic form,
Sometimes such a diffeomorphism is called a symplectiomorphism.
The group of all symplectic diffeomorphisms of
is denoted by
It follows from the nondegeneracy of the symplectic form the map
defines an isomorphism
between the vector fields and the one-forms on a symplectic manifold
If the flow of a vector field
preserves the symplectic form we have that
Then the closedness
of the symplectic form implies that the one-form
is closed. It follows that the Lie algebra of
the group of symplectic diffeomorhism consists of the vector fields
for which the one-form
is closed. Hence it can be identified with the space of closed one-forms.
If the one-form is exact, i.e.
for some function
then the vector field
is called Hamiltonian. Symplectic diffeomorphism generated by Hamiltonian flows form
a group
called the group of Hamiltonian diffeomorphism. Its Lie algebra can be
identified with the quotient of the space of smooth functions on
by the constants.
4 Constructions
4.1 Products
The product of symplectic manifolds and
is a symplectic manifold with
respect to the form
for nonzero real numbers
Here
is the projection.
4.2 Bundles
A locally trivial bundle is called symplectic (resp. Hamiltonian) if its structure
group is a subgroup of the group of symplectic (resp. Hamiltonian) diffeomorphisms.
Example. The product of the Hopf bundle with the circle is a symplectic bundle
Indeed, the structure group is a group of rotations of the torus and hence it preserves the area.
As we have seen above the product
does not admit a symplectic form. This example
shows that, in general, the total space of a symplectic bundle is not symplectic.
Let is a compact symplectic bundle over a symplectic base.
According to a theorem of Thurston, if there exists a cohomology class
such that
its pull back to every fibre is equal to the class of the symplectic form of the fibre
then there exists a representative
of the class
such that
is a symplectic form on
for every big enough
A symplectic fiber bundle may have a symplectic form on the total space which restricts symplectically to the fibers, even if the base is not symplectic. Such bundles are constructed using fat connections. Let there be given a principal fiber bundle
![\displaystyle G\rightarrow P\rightarrow M.](/images/math/4/7/9/47938363919b398e54cca5a5b648eb74.png)
Let be a connection form,
the curvature form of this connection, and
be the horizontal distribution. A vector
is called fat (with respect to the given connection), if the 2-form
![\displaystyle (X,Y)\rightarrow \langle \Omega(X,Y),v\rangle](/images/math/3/4/d/34d3eacec4890082319b1bc6bcc6583f.png)
is nondegenerate on the horizontal distribution of the connection. The following theorem yields a construction of a symplectic form on the total space of fiber bundles associated with principal bundles equipped with particular connections.
Theorem 3.1. Let there be given a symplectic manifold endowed with a hamiltonian action of a Lie group
. Let
be the moment map of the
-action. If
consists of fat vectors, then the associated bundle
![\displaystyle F\rightarrow P\times_GF\rightarrow M](/images/math/d/1/6/d1638f1d5f169a686fbd794fc80e95d7.png)
admits a fiberwise symplectic form on the total space.
Using this theorem, one can construct examples of symplectic fiber bundles with fiberwise symplectic form on the total space (see examples below).
Example (twistor bundles)
Consider the principal bundle of the orthogonal frame bundles over -dimensional manifold
:
![\displaystyle SO(2n)\rightarrow P\rightarrow M.](/images/math/1/e/f/1ef8b4b7a232c6be7b1c1c5442d7c091.png)
Let . The associated bundle with fiber
is called the twistor bundle. It is easy to see that
can be identified with a coadjoint orbit
of some
, where
denotes the Lie algebra of
. Moreover, if
admits Riemannian metric of pinched curvature with sufficiently small pinching constant then
is fat with respect to the Levi-Civitta connection in the frame bundle. As a result, the whole coadjoint orbit (which is the image of the moment map of the
-action) consists of fat vectors. Thus, we obtain a fiberwise symplectic structure on the total space of any twistor bundle
![\displaystyle F_{\xi}\rightarrow P\times_{SO(2n)}F_{\xi}\rightarrow M](/images/math/9/4/b/94bb43209c8a65e171004001385ce792.png)
over even-dimensional manifolds of pinched curvature. In particular, twistor bundles over spheres or hyperbolic manifolds, admit fiberwise symplectic structures. The simplest example of this construction is the fibering of
over
with fiber
, since it is known that the total space of the twistor bundle over
is
.
Example (locally homogeneous complex manifolds)
Let be a Lie group of non-compact type, which is a real form of a complex Lie group
. Choose a parabolic subgroup
and a maximal compact subgroup
in
. Assume that
is compact. Then one can show that
can be identified with a coadjoint orbit of some vector in
, which is fat with respect to a
-invariant connection in the principal bundle
![\displaystyle K\rightarrow G/V\rightarrow G/K.](/images/math/1/b/5/1b5d748be2c73f08dacc0bca71d74b29.png)
It follows that the associated bundle
![\displaystyle K/V\rightarrow G/V\rightarrow G/K](/images/math/e/2/f/e2f51b05a68cf1eeb326e9ecb0ba96cf.png)
is a symplectic fiber bundle with fiberwise symplectic structure. This construction can be compactified by taking lattices in which intersect trivially with
. A particular example is given by the fiber bundle
![\displaystyle SO(2n)/U(n)\rightarrow SO(2n,p)/U(n)\times SO(p)\rightarrow SO(2n,p)/SO(2n)\times SO(p),](/images/math/8/1/f/81f9cb93570d868fb87762952c968555.png)
and its compactification by lattices.
4.3 Symplectic reduction
Let be a Lie group acting on a symplectic manifold
in a hamiltonian way. Denote
by
the moment map of this action. Since
acts on the level set
, one can consider the orbit space
. It is an orbifold in general, but it happens to be a manifold, when
acts freely on the preimage, and
is a regular point. In this case,
is a symplectic manifold as well, called symplectic reduction. It is often denoted by
.
4.4 Symplectic cut
Let be a symplectic manifold with a hamiltonian action of the circle
If
is the moment map,
is a regular level, then the action restricted to
has no fixed points, hence
is the boundary of the associated disk bundle W. This is a manifold if the action is free and an orbifold if a non-trivial isotropy occurs.
4.5 Coadjoint orbits
4.6 Symplectic homogeneous spaces
Nilmanifolds, solvmanifolds, homogeneous spaces of semisimple Lie groups
4.7 Donaldson's theorem on submanifolds
4.8 Surgery in codimension 2
Consider two symplectic manifolds of equal dimension and suppose that there are codimension two symplectic submanifolds
and a symplectomorphism
such that Chern classes of normal bundles satisfy
Then by removing tubular neighborhods of
and
we get manifolds with boundaries. The map
induces a diffeomorphism of the boundaries, one can form a new manifold identifying the boundaries by this diffeomorphism and define on it a symplectic form which coincides with
and
outside of a tubular neigborhood of the trace of glueing [Gompf1995]. The same works if
are symplectic submanifolds of a connected symplectic manifold.
4.9 Symplectic blow-up
5 Invariants
...
6 Classification/Characterization
...
7 Further discussion
...
8 References
- [Gompf1995] R. E. Gompf, A new construction of symplectic manifolds, Ann. of Math. (2) 142 (1995), no.3, 527–595. MR1356781 (96j:57025) Zbl 0849.53027
- [McDuff&Salamon1998] D. McDuff and D. Salamon, Introduction to symplectic topology, The Clarendon Press Oxford University Press, 1998. MR1698616 (2000g:53098) Zbl 0978.53120
![\omega](/images/math/7/2/c/72ca64b196c1c0ec377901f4e543fcd2.png)
![\omega](/images/math/7/2/c/72ca64b196c1c0ec377901f4e543fcd2.png)
![\omega](/images/math/7/2/c/72ca64b196c1c0ec377901f4e543fcd2.png)
![M.](/images/math/b/9/f/b9fe62358899adc4b3b6bc2283ff6608.png)
Symplectic manifolds originated from classical mechanics. The phase space of a dynamical system is the cotangent bundle of the configuration space and it is equipped with a symplectic form. This symplectic form is preserved by the flow of the system.
2 Examples
The most basic example of a symplectic manifold is
equipped with the form
A theorem of Darboux [McDuff&Salamon1998] states that locally every
symplectic manifold if of this form. More precisely, if is a symplectic
-manifold
then for every point
there exists an open neighbourhood
of
and a
diffeomorphism
such that the restriction of
to
is equal to the pull-back
This implies that symplectic manifolds have
no local invariants.
An area form on an oriented surface is symplectic.
Let
be a smooth manifold and let
be a one-form on the cotangent bundle
defined as follows.
If
is a vector tangent to
at a point
then
where
is the projection. In local coordinates the form
can be expressed as
The differential
is a symplectic form on the cotangent bundle
If
is a closed, i.e. compact and without boundary, symplectic
-manifold then the cohomology classes
are non-zero for
This follows from the fact that the cohomology class of the volume
form
is nonzero on a closed manifold. This necessary condition implies that spheres of dimension greater than
two are not symplectic. More generally, no closed manifold of the form
is symplectic for
The complex projective space
is symplectic with respect to its Kähler form.
Its pull back to a complex projective smooth manifold
is also symplectic.
More generally, every Kähler manifold is symplectic.
3 Symmetries
A diffeomorphism of a symplectic manifold
is called symplectic if it preserves
the symplectic form,
Sometimes such a diffeomorphism is called a symplectiomorphism.
The group of all symplectic diffeomorphisms of
is denoted by
It follows from the nondegeneracy of the symplectic form the map
defines an isomorphism
between the vector fields and the one-forms on a symplectic manifold
If the flow of a vector field
preserves the symplectic form we have that
Then the closedness
of the symplectic form implies that the one-form
is closed. It follows that the Lie algebra of
the group of symplectic diffeomorhism consists of the vector fields
for which the one-form
is closed. Hence it can be identified with the space of closed one-forms.
If the one-form is exact, i.e.
for some function
then the vector field
is called Hamiltonian. Symplectic diffeomorphism generated by Hamiltonian flows form
a group
called the group of Hamiltonian diffeomorphism. Its Lie algebra can be
identified with the quotient of the space of smooth functions on
by the constants.
4 Constructions
4.1 Products
The product of symplectic manifolds and
is a symplectic manifold with
respect to the form
for nonzero real numbers
Here
is the projection.
4.2 Bundles
A locally trivial bundle is called symplectic (resp. Hamiltonian) if its structure
group is a subgroup of the group of symplectic (resp. Hamiltonian) diffeomorphisms.
Example. The product of the Hopf bundle with the circle is a symplectic bundle
Indeed, the structure group is a group of rotations of the torus and hence it preserves the area.
As we have seen above the product
does not admit a symplectic form. This example
shows that, in general, the total space of a symplectic bundle is not symplectic.
Let is a compact symplectic bundle over a symplectic base.
According to a theorem of Thurston, if there exists a cohomology class
such that
its pull back to every fibre is equal to the class of the symplectic form of the fibre
then there exists a representative
of the class
such that
is a symplectic form on
for every big enough
A symplectic fiber bundle may have a symplectic form on the total space which restricts symplectically to the fibers, even if the base is not symplectic. Such bundles are constructed using fat connections. Let there be given a principal fiber bundle
![\displaystyle G\rightarrow P\rightarrow M.](/images/math/4/7/9/47938363919b398e54cca5a5b648eb74.png)
Let be a connection form,
the curvature form of this connection, and
be the horizontal distribution. A vector
is called fat (with respect to the given connection), if the 2-form
![\displaystyle (X,Y)\rightarrow \langle \Omega(X,Y),v\rangle](/images/math/3/4/d/34d3eacec4890082319b1bc6bcc6583f.png)
is nondegenerate on the horizontal distribution of the connection. The following theorem yields a construction of a symplectic form on the total space of fiber bundles associated with principal bundles equipped with particular connections.
Theorem 3.1. Let there be given a symplectic manifold endowed with a hamiltonian action of a Lie group
. Let
be the moment map of the
-action. If
consists of fat vectors, then the associated bundle
![\displaystyle F\rightarrow P\times_GF\rightarrow M](/images/math/d/1/6/d1638f1d5f169a686fbd794fc80e95d7.png)
admits a fiberwise symplectic form on the total space.
Using this theorem, one can construct examples of symplectic fiber bundles with fiberwise symplectic form on the total space (see examples below).
Example (twistor bundles)
Consider the principal bundle of the orthogonal frame bundles over -dimensional manifold
:
![\displaystyle SO(2n)\rightarrow P\rightarrow M.](/images/math/1/e/f/1ef8b4b7a232c6be7b1c1c5442d7c091.png)
Let . The associated bundle with fiber
is called the twistor bundle. It is easy to see that
can be identified with a coadjoint orbit
of some
, where
denotes the Lie algebra of
. Moreover, if
admits Riemannian metric of pinched curvature with sufficiently small pinching constant then
is fat with respect to the Levi-Civitta connection in the frame bundle. As a result, the whole coadjoint orbit (which is the image of the moment map of the
-action) consists of fat vectors. Thus, we obtain a fiberwise symplectic structure on the total space of any twistor bundle
![\displaystyle F_{\xi}\rightarrow P\times_{SO(2n)}F_{\xi}\rightarrow M](/images/math/9/4/b/94bb43209c8a65e171004001385ce792.png)
over even-dimensional manifolds of pinched curvature. In particular, twistor bundles over spheres or hyperbolic manifolds, admit fiberwise symplectic structures. The simplest example of this construction is the fibering of
over
with fiber
, since it is known that the total space of the twistor bundle over
is
.
Example (locally homogeneous complex manifolds)
Let be a Lie group of non-compact type, which is a real form of a complex Lie group
. Choose a parabolic subgroup
and a maximal compact subgroup
in
. Assume that
is compact. Then one can show that
can be identified with a coadjoint orbit of some vector in
, which is fat with respect to a
-invariant connection in the principal bundle
![\displaystyle K\rightarrow G/V\rightarrow G/K.](/images/math/1/b/5/1b5d748be2c73f08dacc0bca71d74b29.png)
It follows that the associated bundle
![\displaystyle K/V\rightarrow G/V\rightarrow G/K](/images/math/e/2/f/e2f51b05a68cf1eeb326e9ecb0ba96cf.png)
is a symplectic fiber bundle with fiberwise symplectic structure. This construction can be compactified by taking lattices in which intersect trivially with
. A particular example is given by the fiber bundle
![\displaystyle SO(2n)/U(n)\rightarrow SO(2n,p)/U(n)\times SO(p)\rightarrow SO(2n,p)/SO(2n)\times SO(p),](/images/math/8/1/f/81f9cb93570d868fb87762952c968555.png)
and its compactification by lattices.
4.3 Symplectic reduction
Let be a Lie group acting on a symplectic manifold
in a hamiltonian way. Denote
by
the moment map of this action. Since
acts on the level set
, one can consider the orbit space
. It is an orbifold in general, but it happens to be a manifold, when
acts freely on the preimage, and
is a regular point. In this case,
is a symplectic manifold as well, called symplectic reduction. It is often denoted by
.
4.4 Symplectic cut
Let be a symplectic manifold with a hamiltonian action of the circle
If
is the moment map,
is a regular level, then the action restricted to
has no fixed points, hence
is the boundary of the associated disk bundle W. This is a manifold if the action is free and an orbifold if a non-trivial isotropy occurs.
4.5 Coadjoint orbits
4.6 Symplectic homogeneous spaces
Nilmanifolds, solvmanifolds, homogeneous spaces of semisimple Lie groups
4.7 Donaldson's theorem on submanifolds
4.8 Surgery in codimension 2
Consider two symplectic manifolds of equal dimension and suppose that there are codimension two symplectic submanifolds
and a symplectomorphism
such that Chern classes of normal bundles satisfy
Then by removing tubular neighborhods of
and
we get manifolds with boundaries. The map
induces a diffeomorphism of the boundaries, one can form a new manifold identifying the boundaries by this diffeomorphism and define on it a symplectic form which coincides with
and
outside of a tubular neigborhood of the trace of glueing [Gompf1995]. The same works if
are symplectic submanifolds of a connected symplectic manifold.
4.9 Symplectic blow-up
5 Invariants
...
6 Classification/Characterization
...
7 Further discussion
...
8 References
- [Gompf1995] R. E. Gompf, A new construction of symplectic manifolds, Ann. of Math. (2) 142 (1995), no.3, 527–595. MR1356781 (96j:57025) Zbl 0849.53027
- [McDuff&Salamon1998] D. McDuff and D. Salamon, Introduction to symplectic topology, The Clarendon Press Oxford University Press, 1998. MR1698616 (2000g:53098) Zbl 0978.53120