Indifferential geometry, a subject ofmathematics, asymplectic manifold is asmooth manifold,${\displaystyle M}$, equipped with aclosednondegeneratedifferential 2-form${\displaystyle \omega }$, called the symplectic form. The study of symplectic manifolds is calledsymplectic geometry or symplectic topology. Symplectic manifolds arise naturally in abstract formulations ofclassical mechanics andanalytical mechanics as thecotangent bundles of manifolds. For example, in theHamiltonian 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 thephase space of the system.

Symplectic manifolds arise fromclassical mechanics; in particular, they are a generalization of thephase space of a closed system.^[1] In the same way theHamilton equations allow one to derive the time evolution of a system from a set ofdifferential equations, the symplectic form should allow one to obtain avector field describing the flow of the system from the differential${\displaystyle dH}$ of a Hamiltonian function${\displaystyle H}$.^[2] So we require a linear map${\displaystyle TM\to T^{\ast }M}$ from thetangent manifold${\displaystyle TM}$ to thecotangent manifold${\displaystyle T^{\ast }M}$, or equivalently, an element of${\displaystyle T^{\ast }M\otimes T^{\ast }M}$. Letting${\displaystyle \omega }$ denote asection of${\displaystyle T^{\ast }M\otimes T^{\ast }M}$, the requirement that${\displaystyle \omega }$ benon-degenerate ensures that for every differential${\displaystyle dH}$ there is a unique corresponding vector field${\displaystyle V_H}$ such that${\displaystyle dH=\omega (V_H,\cdot )}$. Since one desires the Hamiltonian to be constant along flow lines, one should have${\displaystyle \omega (V_H,V_H)=dH(V_H)=0}$, which implies that${\displaystyle \omega }$ isalternating and hence a 2-form. Finally, one makes the requirement that${\displaystyle \omega }$ should not change under flow lines, i.e. that theLie derivative of${\displaystyle \omega }$ along${\displaystyle V_H}$ vanishes. ApplyingCartan's formula, this amounts to (here${\displaystyle {\iota }_X}$ is theinterior product):

${\displaystyle {\mathcal{L}}_{V_H}(\omega )=0\iff \mathrm{d}({\iota }_{V_H}\omega )+{\iota }_{V_H}\mathrm{d}\omega =\mathrm{d}(\mathrm{d}H)+\mathrm{d}\omega (V_H)=\mathrm{d}\omega (V_H)=0}$

so that, on repeating this argument for different smooth functions${\displaystyle H}$ such that the corresponding${\displaystyle V_H}$ 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${\displaystyle V_H}$ corresponding to arbitrary smooth${\displaystyle H}$ is equivalent to the requirement thatω should beclosed.

Let${\displaystyle M}$ be a smoothmanifold. Asymplectic form on${\displaystyle M}$ is a closed non-degenerate differential2-form${\displaystyle \omega }$.^[3][4] Here, non-degenerate means that for every point${\displaystyle p\in M}$, the skew-symmetric pairing on thetangent space${\displaystyle T_{p}M}$ defined by${\displaystyle \omega }$ is non-degenerate. That is to say, if there exists an${\displaystyle X\in T_{p}M}$ such that${\displaystyle \omega (X,Y)=0}$ for all${\displaystyle Y\in T_{p}M}$, then${\displaystyle X=0}$. The closed condition means that theexterior derivative of${\displaystyle \omega }$ vanishes.^[3][4]

Asymplectic manifold is a pair${\displaystyle (M,\omega )}$ where${\displaystyle M}$ is a smooth manifold and${\displaystyle \omega }$ is a symplectic form. Assigning a symplectic form to${\displaystyle M}$ is referred to as giving${\displaystyle M}$ asymplectic structure. Since in odd dimensions,skew-symmetric matrices are always singular, nondegeneracy implies that${\displaystyle \dim M}$ is even.

By nondegeneracy,${\displaystyle \omega }$ can be used to define a pair ofmusical isomorphisms${\displaystyle {\omega }^{\mathrm{♭}}:TM\to T^{\ast }M,{\omega }^{\mathrm{♯}}:T^{\ast }M\to TM}$, such that${\displaystyle \omega (X,Y)={\omega }^{\mathrm{♭}}(X)(Y)}$ for any two vector fields${\displaystyle X,Y}$, and${\displaystyle {\omega }^{\mathrm{♯}}\circ {\omega }^{\mathrm{♭}}=\mathrm{Id}}$.

A symplectic manifold${\displaystyle (M,\omega )}$ isexact iff the symplectic form${\displaystyle \omega }$ isexact, i.e. equal to${\displaystyle \omega =-d\theta }$ for some 1-form${\displaystyle \theta }$. The area 2-form on the 2-sphere is an inexact symplectic form, by thehairy ball theorem.

ByDarboux's theorem, around any point${\displaystyle p}$ there exists a local coordinate system, in which${\displaystyle \omega ={\mathrm{\Sigma }}_{i}d{p}_i\wedge d{q}^i}$, where d denotes theexterior derivative and ∧ denotes theexterior product. This form is called thePoincaré two-form or thecanonical two-form. Thus, we can locally think ofM as being thecotangent bundle${\displaystyle T^{\ast }{\mathbb{R}}^n}$ and generated by the correspondingtautological 1-form${\displaystyle \theta ={\mathrm{\Sigma }}_i{p}_{i}d{q}^i,\omega =d\theta }$.

A (local)Liouville form is any (locally defined)${\displaystyle \lambda }$ such that${\displaystyle \omega =d\lambda }$. A vector field${\displaystyle X}$ is (locally)Liouville iff${\displaystyle {\mathcal{L}}_X\omega =\omega }$. ByCartan's magic formula, this is equivalent to${\displaystyle d(\omega (X,\cdot ))=\omega }$. A Liouville vector field can thus be interpreted as a way to recover a (local) Liouville form. By Darboux's theorem, around any point there exists a local Liouville form, though it might not exist globally.

Given any smooth function${\displaystyle f:M\to \mathbb{R}}$, itsHamiltonian vector field is the unique vector field${\displaystyle X_f}$ satisfying${\displaystyle \omega (X_f,\cdot )=df}$. The set of all Hamiltonian vector fields make up aLie algebra, and is written as${\displaystyle (\mathrm{Ham}(M),[\cdot ,\cdot ])}$ where${\displaystyle [\cdot ,\cdot ]}$ is theLie bracket.

Given any two smooth functions${\displaystyle f,g:M\to \mathbb{R}}$, theirPoisson bracket is defined by${\displaystyle \{f,g\}=\omega (X_g,X_f)}$. This makes any symplectic manifold into aPoisson manifold. ThePoisson bivector is abivector field${\displaystyle \pi }$ defined by${\displaystyle \{f,g\}=\pi (df\wedge dg)}$, or equivalently, by${\displaystyle \pi :={\omega }^{-1}}$. The Poisson bracket and Lie bracket are related by$X_{\{f,g\}}=[X_f,X_g]$.

There are several natural geometric notions ofsubmanifold of a symplectic manifold${\displaystyle (M,\omega )}$. Let${\displaystyle N\subset M}$ be a submanifold. It is

• symplectic iff${\displaystyle \omega {|}_N}$ is a symplectic form on${\displaystyle N}$.
• isotropic iff${\displaystyle \omega {|}_N=0}$, equivalently, iff${\displaystyle T_{p}M\subset T_p{M}^{\omega }}$ for any${\displaystyle p\in N}$
• coisotropic iff${\displaystyle T_p{M}^{\omega }\subset T_{p}M}$ for any${\displaystyle p\in N}$.
• Lagrangian iff it is bothisotropic andcoisotropic, i.e.${\displaystyle \omega {|}_L=0}$ and${\displaystyle \text{dim }L=\frac{1}{2}\dim M}$. By the nondegeneracy of${\displaystyle \omega }$, Lagrangian submanifolds are the maximal isotropic submanifolds and minimal coisotropic submanifolds.

The conditions can also be defined bydifferential algebra using Poisson brackets. Let${\displaystyle I_N:=\{f:M\to \mathbb{R}:f{|}_N=0\}}$ be thedifferential ideal of functions vanishing on${\displaystyle N}$, then${\displaystyle N}$ is isotropic iff${\displaystyle \{I_N,I_N\}\subset I_N}$, coisotropic iff${\displaystyle \{I_N,C^{\mathrm{\infty }}(M)\}\subset I_N}$, Lagrangian iff the induced Poisson bracket on the quotient algebra${\displaystyle C^{\mathrm{\infty }}(M)/I_N}$ is zero, and symplectic iff the induced Poisson bracket on the quotient algebra${\displaystyle C^{\mathrm{\infty }}(M)/I_N}$ is nondegenerate.

Lagrangian submanifolds are the most important submanifolds.Weinstein proposed the "symplectic creed":Everything is a Lagrangian submanifold. By that, he means that everything in symplectic geometry is most naturally expressed in terms of Lagrangian submanifolds.^[5]

ALagrangian fibration of a symplectic manifoldM is afibration where all of thefibers are Lagrangian submanifolds.

Given a submanifold${\displaystyle N\subset M}$ of codimension 1, thecharacteristic line distribution on it is the duals to its tangent spaces:${\displaystyle T_p{N}^{\omega }}$. If there also exists a Liouville vector field${\displaystyle X}$ in a neighborhood of it that istransverse to it. In this case, let${\displaystyle \alpha :=\omega (X,\cdot ){|}_N}$, then${\displaystyle (N,\alpha )}$ is acontact manifold, and we say it is acontact type submanifold. In this case, theReeb vector field is tangent to the characteristic line distribution.

Ann-submanifold is locally specified by a smooth function${\displaystyle u:{\mathbb{R}}^n\to M}$. It is a Lagrangian submanifold if${\displaystyle \omega ({\mathrm{\partial }}_i,{\mathrm{\partial }}_j)=0}$ for all${\displaystyle i,j\in 1:n}$. If locally there is a canonical coordinate system${\displaystyle (q,p)}$, then the condition is equivalent to$${\displaystyle [u,v{]}_{p,q}=\sum _{i=1}^n\left(\frac{\mathrm{\partial }{q}_i}{\mathrm{\partial }u}\frac{\mathrm{\partial }{p}_i}{\mathrm{\partial }v}-\frac{\mathrm{\partial }{p}_i}{\mathrm{\partial }u}\frac{\mathrm{\partial }{q}_i}{\mathrm{\partial }v}\right)=0,\mathrm{\forall }i,j\in 1:n}$$where${\displaystyle [\cdot ,\cdot {]}_{p,q}}$ is theLagrange bracket in this coordinate system.

Given any differentiable function${\displaystyle f:M\to \mathbb{R}}$, its differential${\displaystyle df}$ has a graph in${\displaystyle T^{\ast }M}$. The graph is a Lagrangian submanifold. Conversely, if a Lagrangian submanifold${\displaystyle L\subset T^{\ast }M}$ projects down to${\displaystyle M}$ diffeomorphically (i.e. the projection map${\displaystyle \pi :T^{\ast }M\to M}$, when restricted to the submanifold, is a diffeomorphism), then it is the graph of some${\displaystyle df}$ for some${\displaystyle f:M\to \mathbb{R}}$. In such a case,${\displaystyle f}$ is thegenerating function of a Lagrangian manifold.

This example shows that Lagrangian submanifolds satisfy anh-principle, exist in great abundance, and are not rigid. The classification of symplectic manifolds is done viaFloer homology—this is an application ofMorse theory to theaction 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.

LetL be a Lagrangian submanifold of a symplectic manifold (K,ω) given by animmersioni :L ↪K (i is called aLagrangian immersion). Letπ :K ↠B give a Lagrangian fibration ofK. The composite(π ∘i) :L ↪K ↠B is aLagrangian mapping. Thecritical value set ofπ ∘i is called acaustic.

Two Lagrangian maps(π₁ ∘i₁) :L₁ ↪K₁ ↠B₁ and(π₂ ∘i₂) :L₂ ↪K₂ ↠B₂ are calledLagrangian equivalent if there existdiffeomorphismsσ,τ andν such that both sides of the diagram given on the rightcommute, andτ preserves the symplectic form.^[4] Symbolically:

${\displaystyle \tau \circ i_1=i_2\circ \sigma ,\nu \circ {\pi }_1={\pi }_2\circ \tau ,{\tau }^{\ast }{\omega }_2={\omega }_1,}$

whereτ^∗ω₂ denotes thepull back ofω₂ byτ.

A map${\displaystyle f:(M,\omega )\to (M^{\prime },{\omega }^{\prime })}$ between symplectic manifolds is asymplectomorphism when it preserves the symplectic structure, i.e. thepullback is the same${\displaystyle f^{\ast }{\omega }^{\prime }=\omega }$. The most important symplectomorphisms are symplectic flows, i.e. ones generated by integrating a vector field on${\displaystyle (M,\omega )}$.

Given a vector field${\displaystyle X}$ on${\displaystyle (M,\omega )}$, it generates a symplectic flow iff${\displaystyle {\mathcal{L}}_X\omega =0}$. Such vector fields are calledsymplectic. Any Hamiltonian vector field is symplectic, and conversely, any symplectic vector field islocally Hamiltonian.

A property that is preserved under all symplectomorphisms is asymplectic invariant. In the spirit ofErlangen program, symplectic geometry is the study of symplectic invariants.

Let${\displaystyle \{v_1,\dots ,v_{2n}\}}$ be a basis for${\displaystyle {\mathbb{R}}^{2n}.}$ We define our symplectic form${\displaystyle \omega }$ on this basis as follows:

In this case the symplectic form reduces to a simplequadratic form. If${\displaystyle I_n}$ denotes the${\displaystyle n\times n}$identity matrix then the matrix,${\displaystyle \mathrm{\Omega }}$, of this quadratic form is given by the${\displaystyle 2n\times 2n}$block matrix:

That is,

${\displaystyle \omega =\mathrm{d}{x}_1\wedge \mathrm{d}{y}_1+\cdots +\mathrm{d}{x}_n\wedge \mathrm{d}{y}_n.}$

It has a fibration by Lagrangian submanifolds with fixed value of${\displaystyle y}$, i.e.${\displaystyle \{{\mathbb{R}}^n\times \{y\}:y\in {\mathbb{R}}^n\}}$.

A Liouville form for this is$\lambda =\frac{1}{2}\sum _i\left(x_{i}d{y}_i-y_{i}d{x}_i\right)$ and$\omega =d\lambda$, the Liouville vector field is$${\displaystyle Y=\frac{1}{2}\sum _i\left(x_i{\mathrm{\partial }}_{x_i}+y_i{\mathrm{\partial }}_{y_i}\right),}$$the radial field. Another Liouville form is${\displaystyle {\mathrm{\Sigma }}_i{x}_{i}d{y}_i}$, with Liouville vector field$Y=\sum _i{x}_i{\mathrm{\partial }}_{x_i}$.

Let${\displaystyle Q}$ be a smooth manifold of dimension${\displaystyle n}$. Then the total space of thecotangent bundle${\displaystyle T^{\ast }Q}$ has a natural symplectic form, called the Poincaré two-form or thecanonical symplectic form

${\displaystyle \omega =\sum _{i=1}^{n}d{p}_i\wedge d{q}^i}$

Here${\displaystyle (q^1,\dots ,q^n)}$ are any local coordinates on${\displaystyle Q}$ and${\displaystyle (p_1,\dots ,p_n)}$ are fibrewise coordinates with respect to the cotangent vectors${\displaystyle d{q}^1,\dots ,d{q}^n}$. Cotangent bundles are the naturalphase spaces of classical mechanics. The point of distinguishing upper and lower indexes is driven by the case of the manifold having ametric tensor, as is the case forRiemannian manifolds. 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${\displaystyle p_i}$ are "soldered" to the velocities${\displaystyle d{q}^i}$. 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.

The tautological 1-form${\displaystyle \lambda =\sum _i{p}_{i}d{q}^i}$ has Liouville vector field${\displaystyle Y=\sum _i{p}_i{\mathrm{\partial }}_{p_i}}$, the fiberwise radial field. Its flow dilates covectors:$(q,p)\mapsto \left(q,e^{t}p\right)$.

The zero section of the cotangent bundle is Lagrangian. For example, let

${\displaystyle X=\{(x,y)\in {\mathbb{R}}^2:y^2-x=0\}.}$

Then, we can present${\displaystyle T^{\ast }X}$ as

${\displaystyle T^{\ast }X=\{(x,y,\mathrm{d}x,\mathrm{d}y)\in {\mathbb{R}}^4:y^2-x=0,2y\mathrm{d}y-\mathrm{d}x=0\}}$

where we are treating the symbols${\displaystyle \mathrm{d}x,\mathrm{d}y}$ as coordinates of${\displaystyle {\mathbb{R}}^4=T^{\ast }{\mathbb{R}}^2}$. We can consider the subset where the coordinates${\displaystyle \mathrm{d}x=0}$ and${\displaystyle \mathrm{d}y=0}$, giving us the zero section. This example can be repeated for any manifold defined by the vanishing locus of smooth functions${\displaystyle f_1,\dots ,f_k}$ and their differentials${\displaystyle \mathrm{d}{f}_1,\dots ,d{f}_k}$.

AKähler manifold is a symplectic manifold equipped with a compatible integrable complex structure. They form a particular class ofcomplex manifolds. A large class of examples come from complexalgebraic geometry. Any smooth complexprojective variety${\displaystyle V\subset {\mathbb{C}\mathbb{P}}^n}$ has a symplectic form which is the restriction of theFubini—Study form on theprojective space${\displaystyle {\mathbb{C}\mathbb{P}}^n}$.

A symplectic manifold endowed with ametric that iscompatible with the symplectic form is analmost Kähler manifold in the sense that the tangent bundle has analmost complex structure, but this need not beintegrable.

Riemannian manifolds with an${\displaystyle \omega }$-compatiblealmost complex structure are termedalmost-complex manifolds. They generalize Kähler manifolds, in that they need not beintegrable. That is, they do not necessarily arise from a complex structure on the manifold.

The graph of asymplectomorphism in the product symplectic manifold(M ×M,ω × −ω) is Lagrangian. Their intersections display rigidity properties not possessed by smooth manifolds; theArnold conjecture gives the sum of the submanifold'sBetti numbers as a lower bound for the number of self intersections of a smooth Lagrangian submanifold, rather than theEuler characteristic in the smooth case.

In the case ofKähler manifolds (orCalabi–Yau manifolds) we can make a choice${\displaystyle \mathrm{\Omega }={\mathrm{\Omega }}_1+\mathrm{i}{\mathrm{\Omega }}_2}$ on${\displaystyle M}$ as a holomorphic n-form, where${\displaystyle {\mathrm{\Omega }}_1}$ is the real part and${\displaystyle {\mathrm{\Omega }}_2}$ imaginary. A Lagrangian submanifold${\displaystyle L}$ is calledspecial if in addition to the above Lagrangian condition the restriction${\displaystyle {\mathrm{\Omega }}_2}$ to${\displaystyle L}$ is vanishing. In other words, the real part${\displaystyle {\mathrm{\Omega }}_1}$ restricted on${\displaystyle L}$ leads the volume form on${\displaystyle L}$. The following examples are known as special Lagrangian submanifolds,

1. complex Lagrangian submanifolds ofhyperkähler manifolds,
2. fixed points of a real structure of Calabi–Yau manifolds.

InMorse theory, given aMorse function${\displaystyle f:M\to \mathbb{R}}$ and for a small enough${\displaystyle \epsilon }$ one can construct a Lagrangian submanifold given by the vanishing locus${\displaystyle \mathbb{V}(\epsilon \cdot \mathrm{d}f)\subset T^{\ast }M}$. For a generic Morse function we have a Lagrangian intersection given by${\displaystyle M\cap \mathbb{V}(\epsilon \cdot \mathrm{d}f)=\text{Crit}(f)}$.

TheSYZ conjecture deals with the study of special Lagrangian submanifolds inmirror symmetry; see (Hitchin 1999).

TheThomas–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 astability condition on theFukaya category of the manifold.

• Presymplectic manifolds generalize the symplectic manifolds by only requiring${\displaystyle \omega }$ to be closed, but possibly degenerate. Any submanifold of a symplectic manifold inherits a presymplectic structure.
• Poisson manifolds generalize the symplectic manifolds by preserving only thedifferential-algebraic structures of a symplectic manifold.
• Dirac manifolds generalize Poisson manifolds and presymplectic manifolds by preserving even less structure. The definition is designed so that any submanifold of a Poisson manifold induces a Dirac manifold. They can be called "pre-Poisson" manifolds.
• Amultisymplectic manifold of degreek is a manifold equipped with a closed nondegeneratek-form.^[6]
• Apolysymplectic manifold is a Legendre bundle provided with a polysymplectic tangent-valued${\displaystyle (n+2)}$-form; it is utilized inHamiltonian field theory.^[7]

• McDuff, Dusa; Salamon, D. (1998).Introduction to Symplectic Topology. Oxford Mathematical Monographs.ISBN 0-19-850451-9.
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• de Gosson, Maurice A. (2006).Symplectic Geometry and Quantum Mechanics. Basel: Birkhäuser Verlag.ISBN 3-7643-7574-4.
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• Dunin-Barkowski, Petr (2024). "Symplectic duality for topological recursion".Transactions of the American Mathematical Society.arXiv:2206.14792.doi:10.1090/tran/9352.
• "How to find Lagrangian Submanifolds".Stack Exchange. December 17, 2014.
• Lumist, Ü. (2001) [1994],"Symplectic Structure",Encyclopedia of Mathematics,EMS Press
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• McDuff, D. (November 1998)."Symplectic Structures—A New Approach to Geometry"(PDF).Notices of the AMS.
• Hitchin, Nigel (1999). "Lectures on Special Lagrangian Submanifolds".arXiv:math/9907034.

• Differential topology
• Hamiltonian mechanics
• Smooth manifolds
• Symplectic geometry

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