Symplectic manifold

Type of manifold in differential geometry

In differential geometry, a subject of mathematics, a symplectic manifold is a smooth manifold, $M$ , equipped with a closed nondegenerate differential 2-form $\omega$ , 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 bundles of manifolds. For example, in the 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.^[1] In the same way the Hamilton equations allow one to derive the time evolution of a system from a set of differential equations, the symplectic form should allow one to obtain a vector field describing the flow of the system from the differential $dH$ of a Hamiltonian function $H$ .^[2] So we require a linear map $TM\rightarrow T^{*}M$ from the tangent manifold $TM$ to the cotangent manifold $T^{*}M$ , or equivalently, an element of $T^{*}M\otimes T^{*}M$ . Letting $\omega$ denote a section of $T^{*}M\otimes T^{*}M$ , the requirement that $\omega$ be non-degenerate ensures that for every differential $dH$ there is a unique corresponding vector field $V_{H}$ such that $dH=\omega (V_{H},\cdot )$ . Since one desires the Hamiltonian to be constant along flow lines, one should have $\omega (V_{H},V_{H})=dH(V_{H})=0$ , which implies that $\omega$ is alternating and hence a 2-form. Finally, one makes the requirement that $\omega$ should not change under flow lines, i.e. that the Lie derivative of $\omega$ along $V_{H}$ vanishes. Applying Cartan's formula, this amounts to (here $\iota _{X}$ is the interior product):

{\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

so that, on repeating this argument for different smooth functions $H$ such that the corresponding $V_{H}$ 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 $V_{H}$ corresponding to arbitrary smooth $H$ is equivalent to the requirement that ω should be closed.

Definition

Let $M$ be a smooth manifold. A symplectic form on $M$ is a closed non-degenerate differential 2-form $\omega$ .^[3]^[4] Here, non-degenerate means that for every point $p\in M$ , the skew-symmetric pairing on the tangent space $T_{p}M$ defined by $\omega$ is non-degenerate. That is to say, if there exists an $X\in T_{p}M$ such that $\omega (X,Y)=0$ for all $Y\in T_{p}M$ , then $X=0$ . The closed condition means that the exterior derivative of $\omega$ vanishes.^[3]^[4]

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

By nondegeneracy, $\omega$ can be used to define a pair of musical isomorphisms $\omega ^{\flat }:TM\rightarrow T^{*}M,\omega ^{\sharp }:T^{*}M\rightarrow TM$ , such that $\omega (X,Y)=\omega ^{\flat }(X)(Y)$ for any two vector fields $X,Y$ , and $\omega ^{\sharp }\circ \omega ^{\flat }=\operatorname {Id}$ .

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

By Darboux's theorem, around any point $p$ there exists a local coordinate system, in which $\omega =\Sigma _{i}dp_{i}\wedge dq^{i}$ , where d denotes the exterior derivative and ∧ denotes the exterior product. This form is called the Poincaré two-form or the canonical two-form. Thus, we can locally think of M as being the cotangent bundle $T^{*}\mathbb {R} ^{n}$ and generated by the corresponding tautological 1-form $\theta =\Sigma _{i}p_{i}dq^{i},\;\omega =d\theta$ .

A (local) Liouville form is any (locally defined) $\lambda$ such that $\omega =d\lambda$ . A vector field $X$ is (locally) Liouville iff ${\mathcal {L}}_{X}\omega =\omega$ . By Cartan's magic formula, this is equivalent to $d(\omega (X,\cdot ))=\omega$ . 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 $f:M\to \mathbb {R}$ , its Hamiltonian vector field is the unique vector field $X_{f}$ satisfying $\omega (X_{f},\cdot )=df$ . The set of all Hamiltonian vector fields make up a Lie algebra, and is written as $(\operatorname {Ham} (M),[\cdot ,\cdot ])$ where $[\cdot ,\cdot ]$ is the Lie bracket.

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

Submanifolds

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

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

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

Lagrangian submanifolds

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]

A Lagrangian fibration of a symplectic manifold M is a fibration where all of the fibers are Lagrangian submanifolds.

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

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

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

This example shows that Lagrangian submanifolds satisfy an h-principle, exist in great abundance, and are not rigid. The classification of symplectic manifolds is done via Floer homology—this is an application of Morse theory to the 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.

Lagrangian mapping

Let L be a Lagrangian submanifold of a symplectic manifold (K,ω) given by an immersion i : L ↪ K (i is called a Lagrangian immersion). Let π : K ↠ B give a Lagrangian fibration of K. The composite (π ∘ i) : L ↪ K ↠ B is a Lagrangian mapping. The critical value set of π ∘ i is called a caustic.

Two Lagrangian maps (π₁ ∘ i₁) : L₁ ↪ K₁ ↠ B₁ and (π₂ ∘ i₂) : L₂ ↪ K₂ ↠ B₂ are called Lagrangian equivalent if there exist diffeomorphisms σ, τ and ν such that both sides of the diagram given on the right commute, and τ preserves the symplectic form.^[4] Symbolically:

\tau \circ i_{1}=i_{2}\circ \sigma ,\ \nu \circ \pi _{1}=\pi _{2}\circ \tau ,\ \tau ^{*}\omega _{2}=\omega _{1}\,,

where τ^∗ω₂ denotes the pull back of ω₂ by τ.

Symmetries

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

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

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

Examples

The standard symplectic structure

Let $\{v_{1},\ldots ,v_{2n}\}$ be a basis for $\mathbb {R} ^{2n}.$ We define our symplectic form $\omega$ on this basis as follows:

\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}}

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

\Omega ={\begin{pmatrix}0&I_{n}\\-I_{n}&0\end{pmatrix}}.

That is,

\omega =\mathrm {d} x_{1}\wedge \mathrm {d} y_{1}+\dotsb +\mathrm {d} x_{n}\wedge \mathrm {d} y_{n}.

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

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

Cotangent bundles

Let $Q$ be a smooth manifold of dimension $n$ . Then the total space of the cotangent bundle $T^{*}Q$ has a natural symplectic form, called the Poincaré two-form or the canonical symplectic form

\omega =\sum _{i=1}^{n}dp_{i}\wedge dq^{i}

Here $(q^{1},\ldots ,q^{n})$ are any local coordinates on $Q$ and $(p_{1},\ldots ,p_{n})$ are fibrewise coordinates with respect to the cotangent vectors $dq^{1},\ldots ,dq^{n}$ . Cotangent bundles are the natural phase spaces 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 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 $p_{i}$ are "soldered" to the velocities $dq^{i}$ . 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 $\lambda =\sum _{i}p_{i}dq^{i}$ has Liouville vector field $Y=\sum _{i}p_{i}\partial _{p_{i}}$ , the fiberwise radial field. Its flow dilates covectors: ${\textstyle (q,p)\mapsto \left(q,e^{t}p\right)}$ .

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

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

Then, we can present $T^{*}X$ as

T^{*}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 $\mathrm {d} x,\mathrm {d} y$ as coordinates of $\mathbb {R} ^{4}=T^{*}\mathbb {R} ^{2}$ . We can consider the subset where the coordinates $\mathrm {d} x=0$ and $\mathrm {d} y=0$ , giving us the zero section. This example can be repeated for any manifold defined by the vanishing locus of smooth functions $f_{1},\dotsc ,f_{k}$ and their differentials $\mathrm {d} f_{1},\dotsc ,df_{k}$ .

Kähler manifolds

A Kähler manifold is a symplectic manifold equipped with a compatible integrable complex structure. They form a particular class of complex manifolds. A large class of examples come from complex algebraic geometry. Any smooth complex projective variety $V\subset \mathbb {CP} ^{n}$ has a symplectic form which is the restriction of the Fubini—Study form on the projective space $\mathbb {CP} ^{n}$ .

A symplectic manifold endowed with a metric that is 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 integrable.

Almost-complex manifolds

Riemannian manifolds with an $\omega$ -compatible almost complex structure are termed almost-complex manifolds. 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.

Special Lagrangian submanifolds

The graph of a symplectomorphism in the product symplectic manifold (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 numbers as a lower bound for the number of self intersections of a smooth Lagrangian submanifold, rather than the Euler characteristic in the smooth case.

In the case of Kähler manifolds (or Calabi–Yau manifolds) we can make a choice $\Omega =\Omega _{1}+\mathrm {i} \Omega _{2}$ on $M$ as a holomorphic n-form, where $\Omega _{1}$ is the real part and $\Omega _{2}$ imaginary. A Lagrangian submanifold $L$ is called special if in addition to the above Lagrangian condition the restriction $\Omega _{2}$ to $L$ is vanishing. In other words, the real part $\Omega _{1}$ restricted on $L$ leads the volume form on $L$ . The following examples are known as special Lagrangian submanifolds,

complex Lagrangian submanifolds of hyperkähler manifolds,
fixed points of a real structure of Calabi–Yau manifolds.

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

The SYZ conjecture deals with the study of special Lagrangian submanifolds in mirror symmetry; see (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 stability condition on the Fukaya category of the manifold.

Generalizations

Presymplectic manifolds generalize the symplectic manifolds by only requiring $\omega$ 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 the differential-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.
A multisymplectic manifold of degree k is a manifold equipped with a closed nondegenerate k-form.^[6]
A polysymplectic manifold is a Legendre bundle provided with a polysymplectic tangent-valued $(n+2)$ -form; it is utilized in Hamiltonian field theory.^[7]

Citations

^ Webster, Ben (9 January 2012). "What is a symplectic manifold, really?".
^ Cohn, Henry. "Why symplectic geometry is the natural setting for classical mechanics".
^ a b de Gosson, Maurice (2006). Symplectic Geometry and Quantum Mechanics. Basel: Birkhäuser Verlag. p. 10. ISBN 3-7643-7574-4.
^ a b c Arnold, V. I.; Varchenko, A. N.; Gusein-Zade, S. M. (1985). The Classification of Critical Points, Caustics and Wave Fronts: Singularities of Differentiable Maps, Vol 1. Birkhäuser. ISBN 0-8176-3187-9.
^ Weinstein, Alan (1981). "Symplectic geometry". Bulletin of the American Mathematical Society. 5 (1): 1–13. doi:10.1090/S0273-0979-1981-14911-9. ISSN 0273-0979.
^ Cantrijn, F.; Ibort, L. A.; de León, M. (1999). "On the Geometry of Multisymplectic Manifolds". J. Austral. Math. Soc. Ser. A. 66 (3): 303–330. doi:10.1017/S1446788700036636.
^ Giachetta, G.; Mangiarotti, L.; Sardanashvily, G. (1999). "Covariant Hamiltonian equations for field theory". Journal of Physics. A32 (38): 6629–6642. arXiv:hep-th/9904062. Bibcode:1999JPhA...32.6629G. doi:10.1088/0305-4470/32/38/302. S2CID 204899025.

General and cited references

McDuff, Dusa; Salamon, D. (1998). Introduction to Symplectic Topology. Oxford Mathematical Monographs. ISBN 0-19-850451-9.
Hofer, Helmut; Zehnder, Eduard (2011). Symplectic Invariants and Hamiltonian Dynamics. Modern Birkhäuser Classics. Basel: Springer Basel AG Springer e-books. ISBN 978-3-0348-0104-1.
Auroux, Denis. "Seminar on Mirror Symmetry".
Meinrenken, Eckhard. "Symplectic Geometry" (PDF).
Abraham, Ralph; Marsden, Jerrold E. (1978). Foundations of Mechanics. London: Benjamin-Cummings. See Section 3.2. ISBN 0-8053-0102-X.
de Gosson, Maurice A. (2006). Symplectic Geometry and Quantum Mechanics. Basel: Birkhäuser Verlag. ISBN 3-7643-7574-4.
Alan Weinstein (1971). "Symplectic manifolds and their lagrangian submanifolds". Advances in Mathematics. 6 (3): 329–46. doi:10.1016/0001-8708(71)90020-X.
Arnold, V. I. (1990). "Ch.1, Symplectic geometry". Singularities of Caustics and Wave Fronts. Mathematics and Its Applications. Vol. 62. Dordrecht: Springer Netherlands. doi:10.1007/978-94-011-3330-2. ISBN 978-1-4020-0333-2. OCLC 22509804.