Editing Symplectic manifold

{{Short description|Type of manifold in differential geometry}}
{{Use American English|date = March 2019}}

In [[differential geometry]], a subject of [[mathematics]], a '''symplectic manifold''' is a [[Differentiable manifold#Definition|smooth manifold]], <math> M </math>, equipped with a [[Closed and exact differential forms|closed]] [[nondegenerate form|nondegenerate]] [[Differential form|differential 2-form]] <math> \omega </math>, called the symplectic form. The study of symplectic manifolds is called [[symplectic geometry]] or symplectic topology. Symplectic manifolds arise naturally in abstract formulations of [[classical mechanics]] and [[analytical mechanics]] as the [[cotangent bundle]]s of manifolds. For example, in the [[Hamiltonian mechanics|Hamiltonian formulation]] of classical mechanics, which provides one of the major motivations for the field, the set of all possible configurations of a system is modeled as a manifold, and this manifold's cotangent bundle describes the [[phase space]] of the system.

== Motivation ==
Symplectic manifolds arise from [[classical mechanics]]; in particular, they are a generalization of the [[phase space]] of a closed system.<ref name="Webster">{{cite web |first=Ben |last=Webster |title=What is a symplectic manifold, really? |date=9 January 2012 |url=https://sbseminar.wordpress.com/2012/01/09/what-is-a-symplectic-manifold-really/ }}</ref> In the same way the [[Hamilton equations]] allow one to derive the time evolution of a system from a set of [[differential equation]]s, the symplectic form should allow one to obtain a [[vector field]] describing the flow of the system from the differential <math>dH</math> of a Hamiltonian function <math>H</math>.<ref name="Cohn">{{cite web |first=Henry |last=Cohn |title=Why symplectic geometry is the natural setting for classical mechanics |url=https://math.mit.edu/~cohn/Thoughts/symplectic.html }}</ref> So we require a linear map <math>TM \rightarrow T^*M </math> from the [[tangent manifold]] <math>TM</math> to the [[cotangent manifold]] <math> T^* M </math>, or equivalently, an element of <math>T^*M \otimes T^*M</math>. Letting <math>\omega</math> denote a [[Section (fiber bundle)|section]] of <math>T^*M \otimes T^* M</math>, the requirement that <math>\omega</math> be [[Degenerate form|non-degenerate]] ensures that for every differential <math>dH</math> there is a unique corresponding vector field <math>V_H</math> such that <math>dH = \omega (V_H, \cdot)</math>. Since one desires the Hamiltonian to be constant along flow lines, one should have <math>\omega(V_H, V_H) = dH(V_H) = 0</math>, which implies that <math>\omega</math> is [[Alternating form|alternating]] and hence a 2-form. Finally, one makes the requirement that <math>\omega</math> should not change under flow lines, i.e. that the [[Lie derivative]] of <math>\omega</math> along <math>V_H</math> vanishes. Applying [[Cartan homotopy formula|Cartan's formula]], this amounts to (here <math> \iota_X</math> is the [[interior product]]): 

:<math>\mathcal{L}_{V_H}(\omega) = 0\;\Leftrightarrow\;\mathrm d (\iota_{V_H} \omega) + \iota_{V_H} \mathrm d\omega= \mathrm d (\mathrm d\,H) + \mathrm d\omega(V_H) = \mathrm d\omega(V_H)=0</math>

so that, on repeating this argument for different smooth functions <math>H</math> such that the corresponding <math>V_H</math> span the tangent space at each point the argument is applied at, we see that the requirement for the vanishing Lie derivative along flows of <math>V_H</math> corresponding to arbitrary smooth <math>H</math> is equivalent to the requirement that ''ω'' should be [[Closed and exact differential forms|closed]].

== Definition ==
A '''symplectic form''' on a smooth [[manifold]] <math> M </math> is a closed non-degenerate differential [[2-form]] <math> \omega </math>.<ref name="Gosson">{{cite book |first=Maurice |last=de Gosson |title=Symplectic Geometry and Quantum Mechanics |year=2006 |publisher=Birkhäuser Verlag |location=Basel |isbn=3-7643-7574-4 |page=10 }}
</ref><ref name="Arnold">{{Cite book|first1=V. I.|last1=Arnold|first2=A. N.|last2=Varchenko|first3=S. M.|last3=Gusein-Zade|author-link1=Vladimir Arnold|author-link3=Sabir Gusein-Zade|author-link2=Alexander Varchenko|title=The Classification of Critical Points, Caustics and Wave Fronts: Singularities of Differentiable Maps, Vol 1|publisher=Birkhäuser|year=1985|isbn=0-8176-3187-9}}</ref> Here, non-degenerate means that for every point <math> p \in M </math>, the skew-symmetric pairing on the [[tangent space]] <math> T_p M </math> defined by <math> \omega </math> is non-degenerate. That is to say, if there exists an <math> X \in T_p M </math> such that <math> \omega( X, Y ) = 0 </math> for all <math> Y \in T_p M </math>, then <math> X = 0 </math>. Since in odd dimensions, [[skew-symmetric matrices]] are always singular, the requirement that <math> \omega </math> be nondegenerate implies that <math> M </math> has an even dimension.<ref name="Gosson"/><ref name="Arnold"/> The closed condition means that the [[exterior derivative]] of <math> \omega </math> vanishes. A '''symplectic manifold''' is a pair <math> (M, \omega) </math> where <math> M </math> is a smooth manifold and <math> \omega </math> is a symplectic form. Assigning a symplectic form to <math> M </math> is referred to as giving <math> M </math> a '''symplectic structure'''.

== Examples ==
=== Symplectic vector spaces ===
{{main|Symplectic vector space}}

Let <math>\{v_1, \ldots, v_{2n}\}</math> be a basis for <math>\R^{2n}.</math> We define our symplectic form <math>\omega</math> on this basis as follows:

:<math>\omega(v_i, v_j) = \begin{cases} 1 & j-i =n \text{ with } 1 \leqslant i \leqslant n \\ -1 & i-j =n \text{ with } 1 \leqslant j \leqslant n \\ 0 & \text{otherwise} \end{cases}</math>

In this case the symplectic form reduces to a simple [[quadratic form]]. If <math>I_n</math> denotes the <math>n\times n</math> [[identity matrix]] then the matrix, <math>\Omega</math>, of this quadratic form is given by the <math>2n\times 2n</math> [[block matrix]]:

:<math>\Omega = \begin{pmatrix} 0 & I_n  \\ -I_n & 0 \end{pmatrix}. </math>

=== Cotangent bundles ===
Let <math>Q</math> be a smooth manifold of dimension <math>n</math>. Then the total space of the [[cotangent bundle]] <math>T^* Q</math> has a natural symplectic form, called the Poincaré two-form or the [[canonical symplectic form]]

:<math>\omega = \sum_{i=1}^n dp_i \wedge dq^i </math>

Here <math>(q^1, \ldots, q^n)</math> are any local coordinates on <math>Q</math> and <math>(p_1, \ldots, p_n)</math> are fibrewise coordinates with respect to the cotangent vectors <math>dq^1, \ldots, dq^n</math>. Cotangent bundles are the natural [[phase space]]s of classical mechanics.  The point of distinguishing upper and lower indexes is driven by the case of the manifold having a [[metric tensor]], as is the case for [[Riemannian manifold]]s. Upper and lower indexes transform contra and covariantly under a change of coordinate frames.  The phrase "fibrewise coordinates with respect to the cotangent vectors" is meant to convey that the momenta <math>p_i</math> are "[[solder form|soldered]]" to the velocities <math>dq^i</math>. The soldering is an expression of the idea that velocity and momentum are colinear, in that both move in the same direction, and differ by a scale factor.

=== Kähler manifolds ===
A [[Kähler manifold]] is a symplectic manifold equipped with a compatible integrable complex structure. They form a particular class of [[complex manifold]]s. A large class of examples come from complex [[algebraic geometry]]. Any smooth complex [[projective variety]] <math>V \subset \mathbb{CP}^n</math> has a symplectic form which is the restriction of the [[Fubini-Study metric|Fubini—Study form]] on the [[projective space]] <math>\mathbb{CP}^n</math>.

=== Almost-complex manifolds ===
[[Riemannian manifolds]] with an <math>\omega</math>-compatible [[almost complex structure]] are termed [[almost-complex manifold]]s. They generalize Kähler manifolds, in that they need not be [[integrable]].  That is, they do not necessarily arise from a complex structure on the manifold.

== Lagrangian and other submanifolds ==
There are several natural geometric notions of [[submanifold]] of a symplectic manifold <math> (M, \omega) </math>:

* '''Symplectic submanifolds''' of <math> M </math> (potentially of any even dimension) are submanifolds <math> S \subset M </math> such that <math> \omega|_S </math> is a symplectic form on <math> S </math>.
* '''Isotropic submanifolds''' are submanifolds where the symplectic form restricts to zero, i.e. each tangent space is an [[isotropic subspace]] of the ambient manifold's tangent space. Similarly, if each tangent subspace to a submanifold is co-isotropic (the dual of an isotropic subspace), the submanifold is called '''co-isotropic'''.
* '''Lagrangian submanifolds''' of a symplectic manifold <math>(M,\omega)</math> are submanifolds where the restriction of the symplectic form <math>\omega</math> to <math>L\subset M</math> is vanishing, i.e. <math>\omega|_L=0</math> and <math>\text{dim }L=\tfrac{1}{2}\dim M</math>. Lagrangian submanifolds are the maximal isotropic submanifolds.  

One major example is that the graph of a [[symplectomorphism]] in the product symplectic manifold {{nowrap|1=(''M'' × ''M'', ''ω'' × −''ω'')}} is Lagrangian. Their intersections display rigidity properties not possessed by smooth manifolds; the [[Arnold conjecture]] gives the sum of the submanifold's [[Betti number]]s as a lower bound for the number of self intersections of a smooth Lagrangian submanifold, rather than the [[Euler characteristic]] in the smooth case.

=== Examples ===
Let <math>\R^{2n}_{\textbf{x},\textbf{y}}</math> have global coordinates labelled <math>(x_1, \dotsc, x_n, y_1, \dotsc, y_n)</math>. Then, we can equip <math>\R_{\textbf{x},\textbf{y}}^{2n}</math> with the canonical symplectic form 

:<math>\omega =\mathrm{d}x_1\wedge \mathrm{d}y_1 + \dotsb + \mathrm{d}x_n\wedge \mathrm{d}y_n.</math> 

There is a standard Lagrangian submanifold given by <math>\R^n_{\mathbf{x}} \to \R^{2n}_{\mathbf{x},\mathbf{y}}</math>. The form <math>\omega</math> vanishes on <math>\R^n_{\mathbf{x}}</math> because given any pair of tangent vectors <math>X= f_i(\textbf{x}) \partial_{x_i}, Y=g_i(\textbf{x})\partial_{x_i},</math> we have that <math>\omega(X,Y) = 0.</math> To elucidate, consider the case <math>n=1</math>. Then, <math>X = f(x)\partial_x, Y=g(x)\partial_x,</math> and <math>\omega = \mathrm{d}x\wedge \mathrm{d}y</math>. Notice that when we expand this out

:<math>\omega(X,Y) = \omega(f(x)\partial_x,g(x)\partial_x) = \frac{1}{2}f(x)g(x)(\mathrm{d}x(\partial_x)\mathrm{d}y(\partial_x) - \mathrm{d}y(\partial_x)\mathrm{d}x(\partial_x))</math>

both terms we have a <math>\mathrm{d}y(\partial_x)</math> factor, which is 0, by definition.

====Example: Cotangent bundle====
The cotangent bundle of a manifold is locally modeled on a space similar to the first example. It can be shown that we can glue these affine symplectic forms hence this bundle forms a symplectic manifold. A less trivial example of a Lagrangian submanifold is the zero section of the cotangent bundle of a manifold. For example, let 

:<math>X = \{(x,y) \in \R^2 : y^2 - x = 0\}.</math>

Then, we can present <math>T^*X</math> as

:<math>T^*X = \{(x,y,\mathrm{d}x,\mathrm{d}y) \in \R^4 : y^2 - x = 0, 2y\mathrm{d}y - \mathrm{d}x = 0\}</math>

where we are treating the symbols <math>\mathrm{d}x,\mathrm{d}y</math> as coordinates of <math>\R^4 = T^*\R^2</math>. We can consider the subset where the coordinates <math>\mathrm{d}x=0</math> and <math>\mathrm{d}y=0</math>, giving us the zero section. This example can be repeated for any manifold defined by the vanishing locus of smooth functions <math>f_1,\dotsc,f_k</math> and their differentials <math>\mathrm{d}f_1,\dotsc,df_k</math>.

====Example: Parametric submanifold ====
Consider the canonical space <math>\R^{2n}</math> with coordinates <math>(q_1,\dotsc ,q_n,p_1,\dotsc ,p_n)</math>. A parametric submanifold <math>L</math> of <math>\R^{2n}</math> is one that is parameterized by coordinates <math>(u_1,\dotsc,u_n)</math> such that 
:<math>q_i=q_i(u_1,\dotsc,u_n) \quad p_i=p_i(u_1,\dotsc,u_n)</math>

This manifold is a Lagrangian submanifold if the [[Lagrange bracket]] <math>[u_i,u_j]</math> vanishes for all <math>i,j</math>. That is, it is Lagrangian if 

:<math>[u_i,u_j]=\sum_k \frac {\partial q_k}{\partial u_i}\frac {\partial p_k}{\partial u_j} 
- \frac {\partial p_k}{\partial u_i}\frac {\partial q_k}{\partial u_j} 
= 0</math>  

for all <math>i,j</math>.  This can be seen by expanding
:<math>
\frac {\partial }{\partial u_i}=
\frac {\partial q_k}{\partial u_i} \frac {\partial}{\partial q_k} 
+ \frac {\partial p_k}{\partial u_i} \frac {\partial}{\partial p_k} 
</math>
in the condition for a Lagrangian submanifold <math>L</math>. This is that the symplectic form must vanish on the [[tangent manifold]] <math>TL</math>; that is, it must vanish for all tangent vectors:
:<math>\omega\left( \frac {\partial}{\partial u_i}, \frac {\partial}{\partial u_j} \right)=0</math>
for all <math>i,j</math>. Simplify the result by making use of the canonical symplectic form on <math>\R^{2n}</math>:

:<math>
\omega\left( \frac {\partial }{\partial q_k}, \frac {\partial}{\partial p_k}\right) =
-\omega\left( \frac {\partial }{\partial p_k}, \frac {\partial}{\partial q_k}\right) = 1 
 </math>
and all others vanishing.

As [[Chart (topology)|local charts]] on a symplectic manifold take on the canonical form, this example suggests that Lagrangian submanifolds are relatively unconstrained. The classification of symplectic manifolds is done via [[Floer homology]]—this is an application of [[Morse theory]] to the [[Action (physics)|action functional]] for maps between Lagrangian submanifolds. In physics, the action describes the  time evolution of a physical system; here, it can be taken as the description of the dynamics of branes.

====Example: Morse theory====
Another useful class of Lagrangian submanifolds occur in [[Morse theory]]. Given a [[Morse function]] <math>f:M\to\R</math> and for a small enough <math>\varepsilon</math> one can construct a Lagrangian submanifold given by the vanishing locus <math>\mathbb{V}(\varepsilon\cdot \mathrm{d}f) \subset T^*M</math>. For a generic Morse function we have a Lagrangian intersection given by <math>M \cap \mathbb{V}(\varepsilon\cdot \mathrm{d}f) = \text{Crit}(f)</math>.

{{See also|symplectic category}}

=== Special Lagrangian submanifolds ===
In the case of [[Kähler manifold]]s (or [[Calabi–Yau manifolds]]) we can make a choice <math>\Omega=\Omega_1+\mathrm{i}\Omega_2</math> on <math>M</math> as a holomorphic n-form, where <math>\Omega_1</math> is the real part and <math>\Omega_2</math> imaginary. A Lagrangian submanifold <math>L</math> is called '''special''' if in addition to the above Lagrangian condition the restriction <math>\Omega_2</math> to <math>L</math> is vanishing. In other words, the real part <math>\Omega_1</math> restricted on <math>L</math> leads the volume form on <math>L</math>. The following examples are known as special Lagrangian submanifolds,
# complex Lagrangian submanifolds of [[hyperkähler manifold]]s,
# fixed points of a real structure of Calabi–Yau manifolds.
The [[SYZ conjecture]] deals with the study of special Lagrangian submanifolds in [[mirror symmetry (string theory)|mirror symmetry]]; see {{harv|Hitchin|1999}}. 

The [[Thomas–Yau conjecture]] predicts that the existence of a special Lagrangian submanifolds on Calabi–Yau manifolds in Hamiltonian isotopy classes of Lagrangians is equivalent to stability with respect to a [[Bridgeland stability condition|stability condition]] on the [[Fukaya category]] of the manifold.

== Lagrangian fibration ==
A '''Lagrangian fibration''' of a symplectic manifold ''M'' is a [[fibration]] where all of the [[Fiber bundle#Formal definition|fibres]] are Lagrangian submanifolds. Since ''M'' is even-dimensional we can take local coordinates {{nowrap|1=(''p''<sub>1</sub>,...,''p''<sub>''n''</sub>, ''q''<sup>1</sup>,...,''q''<sup>''n''</sup>),}} and by [[Darboux's theorem]] the symplectic form ''ω'' can be, at least locally, written as {{nowrap|1=''ω'' = &sum; d''p''<sub>''k''</sub> &and; d''q''<sup>''k''</sup>}}, where d denotes the [[exterior derivative]] and ∧ denotes the [[exterior product]]. This form is called the [[Poincaré two-form]] or the canonical two-form. Using this set-up we can locally think of ''M'' as being the [[cotangent bundle]] <math>T^*\R^n,</math> and the Lagrangian fibration as the trivial fibration <math>\pi: T^*\R^n \to \R^n.</math> This is the canonical picture.

== Lagrangian mapping ==
[[File:TIKZ PICT FBN.png|thumb|]]
Let ''L'' be a Lagrangian submanifold of a symplectic manifold (''K'',ω) given by an [[Immersion (mathematics)|immersion]] {{nowrap|1=''i'' : ''L'' ↪ ''K''}} (''i'' is called a '''Lagrangian immersion'''). Let {{nowrap|1=''&pi;'' : ''K'' ↠ ''B''}} give a Lagrangian fibration of ''K''. The composite {{nowrap|1=(''&pi;'' ∘ ''i'') : ''L'' ↪ ''K'' ↠ ''B''}} is a '''Lagrangian mapping'''. The '''critical value set''' of ''π'' ∘ ''i'' is called a [[Caustic (mathematics)|caustic]].

Two Lagrangian maps {{nowrap|1=(''&pi;''<sub>1</sub> ∘ ''i''<sub>1</sub>) : ''L''<sub>1</sub> ↪ ''K''<sub>1</sub> ↠ ''B''<sub>1</sub>}} and {{nowrap|1=(''&pi;''<sub>2</sub> ∘ ''i''<sub>2</sub>) : ''L''<sub>2</sub> ↪ ''K''<sub>2</sub> ↠ ''B''<sub>2</sub>}} are called '''Lagrangian equivalent''' if there exist [[diffeomorphism]]s ''σ'', ''τ'' and ''ν'' such that both sides of the diagram given on the right [[commutative diagram|commute]], and ''τ'' preserves the symplectic form.<ref name="Arnold"/> Symbolically:
: <math> \tau \circ  i_1 = i_2 \circ \sigma, \ \nu \circ \pi_1 = \pi_2 \circ \tau, \ \tau^*\omega_2 = \omega_1 \, , </math>
where ''τ''<sup>∗</sup>''ω''<sub>2</sub> denotes the [[Pullback (differential geometry)#Pullback of differential forms|pull back]] of ''ω''<sub>2</sub> by ''τ''.

== Special cases and generalizations ==
* A symplectic manifold <math>(M, \omega)</math> is '''exact''' if the symplectic form <math>\omega</math> is [[closed and exact differential forms|exact]]. For example, the cotangent bundle of a smooth manifold is an exact symplectic manifold if we use the [[canonical symplectic form]].  The area 2-form on the 2-sphere is a symplectic form that is not exact.
* A symplectic manifold endowed with a [[metric tensor|metric]] that is [[Almost complex manifold#Compatible triples|compatible]] with the symplectic form is an [[almost Kähler manifold]] in the sense that the tangent bundle has an [[almost complex structure]], but this need not be [[integrability condition|integrable]].
* Symplectic manifolds are special cases of a [[Poisson manifold]].
* A '''multisymplectic manifold''' of degree ''k'' is a manifold equipped with a closed nondegenerate ''k''-form.<ref>{{cite journal |first1=F. |last1=Cantrijn |first2=L. A. |last2=Ibort |first3=M. |last3=de León |title=On the Geometry of Multisymplectic Manifolds |journal=J. Austral. Math. Soc. |series=Ser. A |volume=66 |year=1999 |issue=3 |pages=303–330 |doi=10.1017/S1446788700036636 |doi-access=free }}</ref>
* A '''polysymplectic manifold''' is a [[Legendre bundle]] provided with a polysymplectic tangent-valued <math>(n+2)</math>-form; it is utilized in [[Hamiltonian field theory]].<ref>{{cite journal |first1=G. |last1=Giachetta |first2=L. |last2=Mangiarotti |first3=G. |last3=Sardanashvily |author-link3=Gennadi Sardanashvily |title=Covariant Hamiltonian equations for field theory |journal=Journal of Physics |volume=A32 |year=1999 |issue=38 |pages=6629–6642 |doi=10.1088/0305-4470/32/38/302 |arxiv=hep-th/9904062 |bibcode=1999JPhA...32.6629G |s2cid=204899025 }}</ref>

== See also ==
{{Portal|Mathematics}}
{{colbegin|colwidth=22em}}
* {{annotated link|Almost symplectic manifold}}
* {{annotated link|Contact manifold}}—an odd-dimensional counterpart of the symplectic manifold.
* {{annotated link|Covariant Hamiltonian field theory}}
* {{annotated link|Fedosov manifold}}
* {{annotated link|Poisson bracket}}
* {{annotated link|Symplectic group}}
* {{annotated link|Symplectic matrix}}
* {{annotated link|Symplectic topology}}
* {{annotated link|Symplectic vector space}}
* {{annotated link|Symplectomorphism}}
* {{annotated link|Tautological one-form}}
* {{annotated link|Wirtinger inequality (2-forms)}}
{{colend}}

== Citations ==
{{Reflist}}

== General and cited references ==
*{{cite book |first1=Dusa |last1=McDuff |author-link=Dusa McDuff |first2=D. |last2=Salamon |title=Introduction to Symplectic Topology |year=1998 |series=Oxford Mathematical Monographs |isbn=0-19-850451-9 }}
*{{cite web |first=Denis |last=Auroux |author-link=Denis Auroux |title=Seminar on Mirror Symmetry |url=https://math.berkeley.edu/~auroux/290s16.html }}
*{{cite web |first=Eckhard |last=Meinrenken |author-link=Eckhard Meinrenken |title=Symplectic Geometry |url=https://www.math.toronto.edu/mein/teaching/LectureNotes/sympl.pdf }}
*{{cite book |first1=Ralph |last1=Abraham |author-link=Ralph Abraham (mathematician) |first2=Jerrold E. |last2=Marsden |author-link2=Jerrold E. Marsden |title=Foundations of Mechanics |year=1978 |publisher=Benjamin-Cummings |location=London |isbn=0-8053-0102-X |at=See Section 3.2 }}
* {{cite book |first=Maurice A. |last=de Gosson |author-link=Maurice A. de Gosson |title=Symplectic Geometry and Quantum Mechanics |year=2006 |publisher=Birkhäuser Verlag |location=Basel |isbn=3-7643-7574-4 }}
* {{cite journal |author=Alan Weinstein |title=Symplectic manifolds and their lagrangian submanifolds |author-link=Alan Weinstein |journal=[[Advances in Mathematics]] |volume=6 |issue=3 |year=1971 |pages=329–46 |doi=10.1016/0001-8708(71)90020-X | doi-access=free }}
* {{Cite book |last=Arnold |first=V. I. |url=http://link.springer.com/10.1007/978-94-011-3330-2 |title=Singularities of Caustics and Wave Fronts |date=1990 |chapter=Ch.1, Symplectic geometry |publisher=Springer Netherlands |isbn=978-1-4020-0333-2 |series=Mathematics and Its Applications |volume=62 |location=Dordrecht |doi=10.1007/978-94-011-3330-2 |oclc=22509804}}

== Further reading  ==
* {{cite journal |first1=Petr |last1=Dunin-Barkowski |year=2024 |title=Symplectic duality for topological recursion |journal=Transactions of the American Mathematical Society |doi=10.1090/tran/9352 |arxiv=2206.14792 }}
* {{cite web |title=How to find Lagrangian Submanifolds |work=[[Stack Exchange]] |date=December 17, 2014 |url=https://math.stackexchange.com/q/1072200 }}
* {{Springer|authorlink=Ülo Lumiste|last=Lumist| first=Ü. |title=Symplectic Structure|id=s/s091860}}
* {{cite journal |author-link=Gennadi Sardanashvily |last=Sardanashvily |first=G. |date=2009 |title=Fibre bundles, jet manifolds and Lagrangian theory |journal=Lectures for Theoreticians |arxiv=0908.1886 }}
* {{cite web |author-link=Dusa McDuff |last=McDuff |first=D. |url=https://www.ams.org/notices/199808/mcduff.pdf |title=Symplectic Structures—A New Approach to Geometry |work=Notices of the AMS |date=November 1998 }}
* {{cite arXiv |first1=Nigel |last1=Hitchin |year=1999 |title=Lectures on Special Lagrangian Submanifolds |eprint=math/9907034 }}

{{Manifolds}}

{{DEFAULTSORT:Symplectic Manifold}}
[[Category:Differential topology]]
[[Category:Hamiltonian mechanics]]
[[Category:Smooth manifolds]]
[[Category:Symplectic geometry]]