Symplectic manifold

A symplectic manifold is an even-dimensional smooth manifold $M$ equipped with a closed nondegenerate differential two-form $\omega$.

A vector field on a symplectic manifold is symplectic when its flow preserves the symplectic form. Infinitesimally this is ${\mathcal{L}}_X\omega =0$. By Cartan's magic formula it is equivalent to $d({\iota }_X\omega )=0$. A Hamiltonian vector field has an exact contraction with the form, so is symplectic. On the standard two-dimensional torus, ${\partial }_x$ contracts to the closed but nonexact form $dy$, providing a symplectic non-Hamiltonian example.

A coisotropic submanifold $N$ satisfies $(T_{p}N{)}^{\omega }\subset T_{p}N$, where the superscript denotes the symplectic orthogonal complement. Every hypersurface of a symplectic manifold is coisotropic.

For a hypersurface $N$ of a symplectic manifold, the restriction of the symplectic form has a one-dimensional kernel, its characteristic line field. On a regular value level $N=H^{-1}(c)$, this line is generated by the Hamiltonian vector field of $H$.

An isotropic submanifold has every tangent space an isotropic subspace of a symplectic vector space. Its dimension is at most half the ambient dimension, and equality makes it a Lagrangian submanifold.

The symplectic form on a $2n$-dimensional symplectic manifold determines the positive volume form ${\omega }^n/n!$ and its total integral, the symplectic volume, using the induced orientation.

The top-degree differential form ${\omega }^n/n!$ on a $2n$-dimensional symplectic manifold. Every canonical transformation preserves this form. For $n>1$, preservation of this volume alone does not imply preservation of the symplectic form.

A symplectic form on a $2n$-dimensional smooth manifold determines the orientation in which the nowhere-zero volume form ${\omega }^n/n!$ is positive. Consequently every symplectic manifold is orientable. In dimension two this forbids a symplectic form on the Möbius strip.

An exact symplectic manifold has a symplectic form that is an exact differential form, written here $\omega =-d\lambda$. A positive-dimensional example without boundary cannot be compact, by the Generalized Stokes theorem applied to the top wedge product of differential forms.

For an exact symplectic manifold with $\omega =d\lambda$, the Liouville vector field is defined by ${\iota }_Z\omega =\lambda$. Cartan's magic formula gives ${\mathcal{L}}_Z\omega =\omega$. On a cotangent bundle using $\omega =-d\alpha$, take $\lambda =-\alpha$; then $Z={\sum }_i{p}_i{\partial }_{p_i}$ generates the cotangent fiber-dilation flow.

The vertical Liouville vector field on a cotangent bundle has complete flow $D_s(x,p)=(x,e^{s}p)$. Its pullbacks scale both the Liouville one-form and the canonical symplectic form by $e^s$. As $s\to -\infty$, each orbit tends to its base point on the zero section. This contraction proves rigidity of cotangent Liouville-form preservation.

On a nonempty compact $2n$-dimensional symplectic manifold without boundary, with $n\ge 1$, the class $[\omega {]}^n$ in top-degree de Rham cohomology is nonzero, since its integral in the symplectic orientation is positive. In particular $\omega$ cannot be an exact differential form: if $\omega =d\alpha$, then ${\omega }^n=d(\alpha \wedge {\omega }^{n-1})$ and the Generalized Stokes theorem would make that integral zero. Thus vanishing second de Rham cohomology obstructs a closed positive-dimensional symplectic manifold.

A symplectic form is a closed differential form of degree two whose value on every tangent space is a nondegenerate alternating bilinear form. In canonical position-momentum coordinates it is ${\sum }_{i}d{q}_i\wedge d{p}_i$.

The standard symplectic form on ${\mathbb{R}}^{2n}$ is

A symplectic ball is a Euclidean ball in $({\mathbb{R}}^{2n},{\omega }_0)$, or its image under a symplectic embedding.

A symplectic embedding $\iota :(N,{\omega }_N)\hookrightarrow (M,{\omega }_M)$ is a smooth embedding satisfying ${\iota }^{\ast }{\omega }_M={\omega }_N$.

The standard symplectic cylinder restricts one conjugate coordinate pair to a disk and leaves the remaining symplectic coordinates unrestricted. Its normalized symplectic capacity is $\pi R^2$, despite its infinite symplectic volume when $n>1$.

If $U$ is nonempty, bounded and open in standard symplectic Euclidean space, and $W$ is a codimension-two symplectic subspace, a linear symplectic change of coordinates makes $W=\{q_1=p_1=0\}$. Boundedness of the remaining coordinate projection places $U+W$ inside a finite-radius symplectic cylinder. Openness puts a small ball inside it. Monotonicity and normalization of a symplectic capacity give the displayed strict bounds.

A symplectic blowup at a point removes a small symplectic ball and collapses its Hopf-circle boundary fibers, replacing the center by an exceptional divisor $E\cong {\mathbb{CP}}^{n-1}$. The resulting form agrees with the original form away from the surgery region, and its integral over a projective line in $E$ records the blowup size. In real dimension four, $E\cong {\mathbb{CP}}^1$ and $E^2=-1$.

For a symplectic blowup of a $2n$-dimensional symplectic manifold, the displayed formula gives its First Chern class. The exceptional divisor has normal bundle $\mathcal{O}(-1)$; the discrepancy coefficient is $n-1$. In dimension four it is one.

For a closed $2n$-dimensional symplectic manifold, a symplectic blowup of size $\epsilon =\pi R^2$ removes the symplectic volume of the radius-$R$ ball. Its exceptional divisor has lower dimension and contributes no ambient volume. The displayed volume loss shows that different size choices can give forms that are not related by a symplectomorphism.

The Darboux theorem says that every point of a $2n$-dimensional symplectic manifold has local coordinates in which the symplectic form is ${\sum }_{j}d{x}_j\wedge d{y}_j$. Thus symplectic manifolds have no local invariants beyond dimension.

A Darboux chart is a coordinate chart in which a symplectic form becomes the standard symplectic form.

On a closed oriented smooth four-manifold $X$, a Lefschetz pencil is a smooth map $f:X\setminus B\to {\mathbb{CP}}^1$ with finite base locus $B$ and isolated critical points. Orientation-compatible complex local coordinates give the modelsA regular fiber closes to a surface through the base points. Blowing up those points resolves the local ratio and gives a Lefschetz fibration, with an exceptional section for each base point.^[ref]

For a pencil generated by homogeneous polynomials $F_0,F_1$ on a smooth projective surface, $f=[F_0:F_1]$ and $B$ is their common zero set. Generic choices in sufficiently positive linear systems give the isolated complex Morse singularities required here. The local models are stronger conditions than merely having a rational map to the projective line.

Let $X$ be a smooth projective surface over the complex numbers, and let $H$ be an ample divisor. For sufficiently large integers $m$, a general pencil in $|mH|$ is a Lefschetz pencil. Its smooth fiber represents $mH$, so the arithmetic adjunction formula on a smooth surface givesIf $m=4k$, then $m^2{H}^2/2=8k^2{H}^2$ and $m{K}_X\cdot H/2=2k{K}_X\cdot H$ are both even integers. Therefore $g$ is odd. Positivity is needed to obtain a generic Lefschetz pencil; divisibility by four is the separate parity argument. The genericity assertion here uses complex algebraic geometry, rather than an unqualified positive-characteristic version of the complex Morse local model.^[ref]

The standard four-torus has trivial complex tangent bundle, so its symplectic canonical class is zero. The symplectic adjunction formula gives $F^2=2g-2$ for a smooth genus-$g$ fiber of a symplectic Lefschetz pencil. Two generic fibers meet once positively at each base point and nowhere else, giving the displayed count.

A Lefschetz fibration is a proper map from an oriented smooth manifold $X^{2n}$ to an oriented surface, with isolated critical points modeled in orientation-compatible complex coordinates byAway from the critical values it is a proper submersion, hence a smooth fiber bundle by the Ehresmann fibration theorem. Properness is needed for this conclusion; the local critical-point model alone cannot ensure that fibers do not escape to infinity.^[ref]

In real dimension four, a regular fiber is a compact surface. Near a critical value, a circle in that surface shrinks to a node. Blowing up the base points of a Lefschetz pencil supplies a central family of examples: each exceptional sphere becomes a section of the resolved fibration.

If a Lefschetz pencil on a closed four-manifold $X$ has smooth fiber of genus $g$, $b$ base points and $m$ critical points, thenIndeed, blowing up the base points gives $\chi (\tilde{X})=\chi (X)+b$, while a genus-$g$ fibration over $S^2$ contributes $2(2-2g)$ and each Lefschetz critical point contributes one.

Every symplectic smooth fiber in a Lefschetz pencil on ${\mathbb{CP}}^2$ represents $dH$ for an integer $d\ge 1$. The symplectic adjunction formula givesConversely, generic pencils of degree-$d$ complex plane curves realize every value in this list.

A symplectic surface is a two-dimensional symplectic manifold. Every embedded curve in it is a Lagrangian submanifold.

The standard symplectic plane is ${\mathbb{R}}^2$ with the closed nondegenerate two-form $\omega =dq\wedge dp$. For tangent vectors $u=(a,b)$ and $v=(c,d)$,their signed parallelogram area. This alternating pairing is not a positive-definite inner product: every vector pairs to zero with itself.

A linear map $A$ preserves the form exactly when $\det A=1$, because $\omega (Au,Av)=\det (A)\omega (u,v)$. Thus $\mathrm{Sp}(2,\mathbb{R})={\mathrm{SL}}_2(\mathbb{R})$. A nonlinear example is the shear $(q,p)\mapsto (q,p+q^2)$, whose pullback form is $dq\wedge (dp+2qdq)=dq\wedge dp$.

Every smooth embedded curve in the plane is Lagrangian: the restricted two-form vanishes on its one-dimensional tangent spaces, which have half the ambient dimension. The symplectic plane is therefore the simplest model of symplectic area preservation and Lagrangian geometry.

The symplectic area of a compact symplectic surface $(\Sigma ,\omega )$ is ${\int }_{\Sigma }\omega$, using the orientation determined by the symplectic form.

Two closed, connected symplectic surfaces are symplectomorphic exactly when their genus of a surface and total symplectic area agree. Necessity follows from invariance of topology and the integral of the pulled-back symplectic form.

For sufficiency, the classification of compact surfaces supplies an orientation-preserving diffeomorphism between surfaces of the same genus. Pull both forms back to one surface. Their equal integrals make their classes in $H^2(\Sigma ;\mathbb{R})$ equal. The interpolation ${\omega }_t=(1-t){\omega }_0+t{\omega }_1$ remains positive, since in two dimensions the two positive forms are pointwise positive multiples of each other. Moser theorem now gives an isotopy identifying them.^[ref]

The boundaryless hypothesis matters. A disk and an annulus can have the same genus and the same total area, but different numbers of boundary components prevent even a diffeomorphism. Boundary versions of the classification require the corresponding topology and boundary conditions.

A symplectomorphism of a closed symplectic surface carries the complementary components of a separating simple closed curve to those of its image, preserving their unordered symplectic areas. On the unit sphere a latitude at height $h$ has complementary areas $2\pi (1-h)$ and $2\pi (1+h)$. Thus the equator cannot be sent to any nonequatorial latitude.

A symplectomorphism $\varphi :(M,\omega )\to (M^{\prime },{\omega }^{\prime })$ is a diffeomorphism satisfying ${\varphi }^{\ast }{\omega }^{\prime }=\omega$.

This group consists of time-one maps of Hamiltonian isotopies whose flows are defined throughout the specified time interval. With ${\iota }_{X_H}\omega =-dH$, composition of isotopies generated by $H_t,K_t$ has generator $H_t+K_t\circ {\varphi }_t^{-1}$, where ${\varphi }_t$ is the first isotopy. The inverse path has generator $-H_t\circ {\varphi }_t$. These formulas prove closure under composition and inversion. On a closed symplectic manifold, every smooth time-dependent Hamiltonian has a complete flow over a bounded time interval.

This subgroup uses Hamiltonians whose support in $[0,1]\times M$ is compact. Their flows are the identity outside a fixed compact set. On closed manifolds this support condition imposes no extra restriction. On an open symplectic manifold, it excludes nontrivial whole-fiber cotangent fiber translations, even when their generating one-form is exact.

The displayed distance uses the oscillation length of a compactly supported Hamiltonian isotopy. On a closed manifold one may normalize each Hamiltonian to have zero mean. Additive time-dependent constants do not change the flow or its oscillation. Pullback by a symplectomorphism preserves extrema, so this distance is bi-invariant. Nondegeneracy requires an additional geometric argument; it does not follow merely from the definition as an infimum of path lengths.

Assume the lower bound $e(L)\ge \gamma (L)/2$ for a closed rational Lagrangian submanifold, with matching energy convention. A nonidentity Hamiltonian map displaces a sufficiently small ball about some moved point. Inside that ball place a Lagrangian torus formed from equal-area circles. Its least positive Liouville period is positive, so the map's norm is at least half this period. This proves positivity of the distance between different maps without appealing to nondegeneracy itself.

The displacement energy of a subset of a symplectic manifold is the infimum of the energy norms of compactly supported Hamiltonian diffeomorphisms moving it off itself. The infimum is $+\infty$ if no such map exists. One must use the same chosen norm convention in both this definition and any energy lower bound. A compact set can have zero displacement energy even when every nonidentity map has positive norm.

Use the standard symplectic form ${\sum }_{j}d{q}_j\wedge d{p}_j$ and convention ${\iota }_{X_H}\omega =-dH$. The Hamiltonian $H=\delta q_n$ translates $p_n$ by $\delta$. A fixed compactly supported cutoff equal to one near the set swept for $|\delta |\le 1$ leaves that translation unchanged on the original compact set. Its supremum norm is at most $C|\delta |$, so the translation displaces the set at arbitrarily small cost. This proves zero Hamiltonian displacement energy for either Hofer normalization.

Use compactly supported Hamiltonians on open manifolds and zero-mean Hamiltonians on closed manifolds. Because their values straddle zero, $\Vert H_t{\Vert }_{\infty }\le \max H_t-\min H_t\le 2\Vert H_t{\Vert }_{\infty }$. Thus ${\rho }_{\infty }\le d_H\le 2{\rho }_{\infty }$. These are two equivalent, but not identically normalized, conventions for Hofer's metric. A statement about zero distance or zero displacement energy is independent of this choice.

A symplectic isotopy is a smooth path ${\varphi }_t$ of symplectomorphisms beginning at the identity.

A strong isotopy of symplectic forms is a smooth path of symplectic forms whose de Rham cohomology class is constant, so that Moser's method can seek an ambient isotopy relating its endpoints.

For a symplectic isotopy generated by $X_t$, its flux isIt vanishes for a Hamiltonian isotopy.

For a smooth function $H_t:M\to \mathbb{R}$, the Hamiltonian vector field is defined by ${\iota }_{X_{H_t}}\omega =-d{H}_t$, with the sign depending on convention.

Compactly supported Hamiltonian vector fields can attain any prescribed tangent vector at a point of a symplectic manifold: prescribe the derivative of a local Hamiltonian and multiply by a bump function. Composing flows for a tangent basis gives an open orbit by the inverse function theorem. Every orbit is open, so on a connected manifold the compactly supported Hamiltonian transformations act transitively.

On a symplectic manifold, use ${\iota }_{X_f}\omega =-df$ and $\{f,g\}=\omega (X_f,X_g)=X_f(g)$ for the Poisson bracket. Cartan's magic formula gives ${\mathcal{L}}_{X_f}\omega =0$. Taking the interior product of a differential form with $[X_f,X_g]$ then yields $-d\{f,g\}$, proving the homomorphism. Its kernel consists of locally constant functions, since $X_f=0$ precisely when $df=0$. On a connected manifold these are the constants. The quotient of smooth functions by this kernel is therefore the Lie algebra of Hamiltonian vector fields. A different contraction sign requires a compatible bracket sign.

For independent integrals $G_1=H,G_2,\dots ,G_r$, the Hamiltonian $K=H+\frac{1}{2}{\sum }_j{G}_j^2$ agrees with $H$ to first order on the common zero level. On a nonempty level $G=c\ne 0$, the vector-field difference is ${\sum }_j{c}_j{X}_{G_j}$ and is nowhere zero by independence of the differentials. Nonemptiness is necessary if one expresses the conclusion as inequality between restricted vector fields.

A Hamiltonian function is a smooth function whose differential determines a Hamiltonian vector field.

A Hamiltonian isotopy is a smooth path of symplectomorphisms ${\varphi }_t$, with ${\varphi }_0=\mathrm{id}$, generated by a smooth time-dependent Hamiltonian function. In the convention ${\iota }_{X_{H_t}}\omega =-d{H}_t$, it satisfiesThe flow must exist throughout the chosen time interval. This is automatic on a closed symplectic manifold; on an open one, uniformly compactly supported Hamiltonians also suffice. The Hamiltonian flow preserves the symplectic form by Cartan's magic formula.

Its time-one map is a Hamiltonian diffeomorphism, an element of the group of Hamiltonian diffeomorphisms. A path and its endpoint are different data: a full rotation of the symplectic plane gives a nonconstant Hamiltonian path whose endpoint is the identity. Distinct paths can therefore represent the same Hamiltonian diffeomorphism.

A Hamiltonian diffeomorphism is the time-one map of a Hamiltonian isotopy beginning at the identity on a symplectic manifold. The isotopy is generated by a smooth time-dependent Hamiltonian function $H_t$. Its flow must exist for the whole interval; on an open manifold a common group convention requires compact support throughout that interval.

In the convention ${\iota }_{X_{H_t}}\omega =-d{H}_t$, the infinitesimal preservation law isThus every Hamiltonian diffeomorphism is a symplectomorphism. The conclusion follows from closedness of the symplectic form, not from a special choice of coordinates.

The identity is generated by the zero Hamiltonian. Inverses and compositions are again Hamiltonian endpoints, forming the group of Hamiltonian diffeomorphisms. A generating function path and its endpoint are different data: different isotopies can have the same time-one map. The Hamiltonian condition imposes more structure than only preserving the symplectic form.

If ${\iota }_{X_{H_t}}\omega =-d{H}_t$ and ${\varphi }_t$ is its flow, then Cartan's magic formula givesThus every defined time slice of a Hamiltonian isotopy is a symplectomorphism.

Given a bounded set in the symplectic plane and a translation vector $v=(a,b)$, the linear Hamiltonian $H(x,y)=bx-ay$ generates translation by $v$. Multiplying it by a cutoff that equals one on the entire swept region produces a compactly supported Hamiltonian diffeomorphism that agrees with the translation on the set.

A Lagrangian submanifold of a $2n$-dimensional symplectic manifold is an $n$-dimensional submanifold $L$ satisfying $\omega {|}_L=0$. The symplectic form identifies its normal bundle with its cotangent bundle.

The graph of a one-form $\alpha$ on a smooth manifold $Q$ is a Lagrangian submanifold of its cotangent bundle exactly when $d\alpha =0$. Indeed the graph pulls the tautological one-form back to $\alpha$, and therefore pulls the canonical symplectic form back to $-d\alpha$. Locally the Poincaré lemma writes $\alpha =dS$, giving $p_i={\partial }_{i}S$. Globally a closed one-form may have nonzero periods and no single generating function.

A foliation of a symplectic manifold is Lagrangian when every leaf has half the ambient dimension and the restricted symplectic form vanishes. The tangent distribution is a real polarization in geometric quantization. Restricted prequantum connections are flat on individual leaves, but nontrivial holonomy can obstruct global parallel sections.

A leaf of a Lagrangian foliation is an immersed Lagrangian submanifold tangent to its integrable distribution. A prequantum line bundle restricts to a flat connection along such a leaf, because its curvature is proportional to the symplectic form. Nonzero global parallel sections require trivial holonomy along every leaf loop.

A Lagrangian submanifold in a Calabi–Yau geometry is special when the phase of its holomorphic volume form is constant. With orientation chosen so that the real part is positive, it is a calibrated submanifold for $\mathrm{Re}(e^{-i\theta }\Omega )$. In ${\mathbb{C}}^3$, its spatial three-fold can describe a supersymmetric curved D3-brane.

For a graph $z=x+i\nabla f$ in ${\mathbb{C}}^3$, symmetry of the Hessian matrix makes the graph Lagrangian. Pulling back $\Omega =d{z}^1\wedge d{z}^2\wedge d{z}^3$ gives $\det (I+i{D}^{2}f)$. Its imaginary part is $\mathrm{tr}{D}^{2}f-\det D^{2}f$, while its real part is $1-{\sigma }_2(D^{2}f)$. Vanishing of the imaginary part and positivity of the real part give the displayed equation and oriented branch. The equation is nonlinear even though its linearization near a flat graph is the Laplace equation.

// Target: string-theory.bigb

For a smooth function $M(q)$, its differential defines a Lagrangian submanifold of the cotangent bundle: pullback of the canonical one-form on a cotangent bundle is $dM$, so pullback of the symplectic form $-d\theta$ is zero. The graph has half the ambient dimension. With $q=(S,V,N)$ and momentum coordinates $(T,-P,\mu )$, the first law of thermodynamics identifies the equilibrium graph with $dM$. This geometric statement does not require extensivity or thermodynamic stability.

A Lagrangian submanifold diffeomorphic to an $n$-dimensional flat torus in a $2n$-dimensional symplectic manifold. Compact connected regular levels of a completely integrable Hamiltonian system are examples, by the Arnold-Liouville theorem.

Choose a reference path between Lagrangian submanifolds $L_0,L_1$ in a symplectic manifold. With a connecting homotopy strip oriented by $ds\wedge dt$, define $\mathcal{A}(\gamma )=-\int u^{\ast }\omega$. If the path space is connected and $H_2(M,L_0\cup L_1)=0$, relative symplectic-area independence makes this single-valued up to a global constant. Its derivative is $d\mathcal{A}(\xi )=\int \omega (\dot{\gamma },\xi )dt$, so its critical points are the constant paths in $L_0\cap L_1$. For a compatible almost complex structure, its formal downward $L^2$ flow is the J-holomorphic curve equation $u_s+J{u}_t=0$.

Two homotopy strips with the same endpoint paths and boundary on Lagrangian submanifolds glue to a relative $2$-cycle. If that cycle is zero in $H_2(M,L_0\cup L_1)$, its symplectic area vanishes: apply Stokes theorem, using $d\omega =0$ and the zero restriction of $\omega$ to each Lagrangian. Thus relative homology controls the ambiguity in a Lagrangian path-area functional.

A Lagrangian embedding is a smooth embedding whose image is a Lagrangian submanifold.

A Lagrangian immersion is an immersion $\iota :L^n\to (M^{2n},\omega )$ satisfying ${\iota }^{\ast }\omega =0$.

If ${\gamma }_i:L_i\to (M_i,{\omega }_i)$ are one-dimensional immersed curves in symplectic surfaces, thenis a Lagrangian immersion. A double point of one factor times the other curve becomes a clean self-intersection circle.

A clean Lagrangian intersection is a clean intersection between two local Lagrangian sheets. A small exact perturbation of one sheet near a compact clean component replaces the intersection by the critical points of a Morse function on that component.

Lagrangian surgery replaces two transverse local Lagrangian sheets by a smooth Lagrangian neck. For the standard planes ${\mathbb{R}}^n$ and $i{\mathbb{R}}^n$ in ${\mathbb{C}}^n$, a neck is obtained by following a curve between the two coordinate rays and multiplying it by points of $S^{n-1}$. In dimension four, resolving a double point attaches a one-handle to the source surface and lowers its Euler characteristic by two.

If two embedded Lagrangian surfaces meet transversely in one point, Lagrangian surgery at that point produces their connected sum. For orientable surfaces ${\Sigma }_{g_1}$ and ${\Sigma }_{g_2}$, the result is ${\Sigma }_{g_1+g_2}$.

Perturb a clean self-intersection circle of a Lagrangian immersion by a Morse function on $S^1$. Its minimum and maximum give two transverse double points. Resolving both by Lagrangian surgery adds two one-handles and changes the Euler characteristic by $-4$. One neck may be glued with the orientation-reversing choice, so the resolved surface is nonorientable.

For every $l\ge 1$, take a plane immersed circle with $l$ transverse double points and form its product with an embedded circle in a second symplectic plane. The resulting torus is a Product Lagrangian immersion with $l$ clean double circles. Resolving them gives a connected nonorientable Lagrangian embedding in ${\mathbb{R}}^4$ with Euler characteristic $-4l$.

The cotangent bundle has the canonical one-form ${\lambda }_{(q,p)}(v)=p(d\pi (v))$ and the canonical symplectic form $d\lambda$ up to a conventional sign. Its zero section is a Lagrangian submanifold.

On a Riemann surface, a meromorphic n-differential is a meromorphic section of the $n$th power of the canonical bundle. In a coordinate it is $F(z)(dz{)}^n$, with coefficients transforming by the $n$th power of the derivative. At a ramification coordinate $w=t^e$, its pullback has order $e{\mathrm{ord}}_w\omega +n(e-1)$. At a modular cusp, the exponential coordinate gives $dz=(h/(2\pi i))dq/q$, so a weight-$2n$ form of cusp order $v$ produces a differential of order $v-n$.

A differential one-form $\theta$ is a smooth section of a vector bundle $T^{\ast }X\to X$. Its graph is its image as a section, and the projection restricts to a diffeomorphism onto $X$. Pullback of the Liouville one-form by that section equals $\theta$, making the twisted Lagrangian graph criterion immediate.

Every cotangent bundle $T^{\ast }M$ is an orientable smooth manifold even when its base is not. The intrinsic canonical one-form on a cotangent bundle ${\lambda }_{(p,\alpha )}(W)=\alpha (d\pi W)$ has $d\lambda ={\sum }_{i}d{a}_i\wedge d{x}^i$. Its $n$th wedge power is the nowhere-zero form $n!d{a}_1\wedge d{x}^1\wedge \cdots \wedge d{a}_n\wedge d{x}^n$, which defines an orientation of the $2n$-dimensional total space. Using $-d\lambda$ gives another standard sign convention; either is a symplectic form.

For coordinates $x,y$ on a smooth manifold, fibre coefficients of a cotangent space element obey $b_j={\sum }_i{a}_i\partial x^i/\partial y^j$. Thus induced cotangent bundle charts change by $(x,a)\mapsto (y(x),b)$ with smooth, invertible, fibre-linear changes. The canonical one-form on a cotangent bundle satisfies ${\sum }_i{a}_{i}d{x}^i={\sum }_j{b}_{j}d{y}^j$, and differentiating proves invariance of its exterior derivative.

For the cotangent bundle projection $\pi :T^{\ast }X\to X$, the canonical differential one-form is ${\lambda }_{(x,\xi )}(v)=\xi (d\pi (v))$. In coordinates it is ${\sum }_j{\xi }_{j}d{x}_j$. The canonical symplectic form is $-d\lambda$ in the position-first convention ${\sum }_{j}d{x}_j\wedge d{\xi }_j$.

Translation by a differential one-form $\beta$ is $(x,\xi )\mapsto (x,\xi +{\beta }_x)$. It changes the canonical one-form on a cotangent bundle by ${\pi }^{\ast }\beta$. A closed differential form gives a symplectomorphism; an exact differential form gives a Hamiltonian diffeomorphism when compact support is not required.

If $dh$ is not identically zero, translating cotangent fibers by $dh$ preserves the canonical symplectic form but changes the Liouville one-form by ${\pi }^{\ast }dh$. Any positive-dimensional smooth manifold has a suitable nonconstant bump function in a chart. In dimension zero every differential one-form is zero, so this distinction cannot occur.

For a diffeomorphism $f:X\to X$, its cotangent lift isIt preserves the canonical one-form and therefore the canonical symplectic form. Cotangent lift is functorial under composition.

If $x=\chi (x^{\prime })$ is an angular diffeomorphism and $\eta \cdot dx$ is a closed differential one-form, the map $y=\eta (\chi (x^{\prime }))+D\chi (x^{\prime }{)}^{-T}{y}^{\prime }$ is a symplectomorphism. It combines a cotangent lift of a diffeomorphism with translation by a closed differential one-form. In particular $\eta =du+a$ is allowed on a torus even when the constant one-form $a\cdot dx$ is not exact.

On either branch $q_1\ne 0$, all local canonical momenta for these coordinates have the form $P_1=(p_1-p_2)/(2q_1)+{\partial }_{Q_1}S$ and $P_2=p_2+{\partial }_{Q_2}S$, with arbitrary smooth $S(Q_1,Q_2)$. The mixed coordinate–momentum brackets fix the linear momentum terms, while the momentum–momentum bracket requires the remaining shift one-form to be closed. It is locally a gradient. The transformation is singular at $q_1=0$ and is not globally one-to-one across both signs of $q_1$.

A diffeomorphism of the entire cotangent bundle preserving the Liouville one-form is the cotangent lift of a diffeomorphism of the base. It preserves the zero section and the Liouville vector field, hence commutes with the cotangent fiber-dilation flow. The limit as the fiber contracts forces it to cover its restriction to the zero section. Evaluation of the preserved one-form then forces the inverse-transpose derivative action on each covector.

For a submanifold $Y\subseteq X$, its conormal bundle consists of covectors annihilating $TY$. A cotangent lift sends $N^{\ast }Y$ to $N^{\ast }f(Y)$.

If $\sigma$ is a closed two-form on $X$, then $\omega +{\pi }^{\ast }\sigma$ is a symplectic form on $T^{\ast }X$, called a twisted cotangent symplectic form. Nondegeneracy follows from the horizontal-vertical block form of the canonical symplectic form.

If $\sigma =d\beta$, cotangent fiber translation by $+\beta$ gives a symplectomorphism from $(T^{\ast }X,-d\alpha )$ to $(T^{\ast }X,-d\alpha +{\pi }^{\ast }\sigma )$. Its pullback changes $\alpha$ by ${\pi }^{\ast }\beta$, so the extra two-form cancels with $-{\pi }^{\ast }d\beta$. The opposite translation gives the inverse direction, globally without compactness.

With $\omega =-d\alpha$ and ${\omega }_{\sigma }=\omega +{\pi }^{\ast }\sigma$, the one-form graph of $\theta$ is a Lagrangian submanifold exactly when $d\theta =\sigma$. Indeed its pulled-back symplectic form is $\sigma -d\theta$ and its dimension is half that of the total space. This extends graph of a closed one-form is Lagrangian.

If $[\sigma ]\ne 0$ in de Rham cohomology, a twisted cotangent symplectic form has no Lagrangian submanifold whose projection is a diffeomorphism onto the entire base. Such a submanifold would be a global one-form graph, and the twisted Lagrangian graph criterion would make $\sigma$ exact.

The graph of a one-form $\beta$ in $T^{\ast }X$ is Lagrangian exactly when $d\beta =0$. Its intersections with the zero section are the zeros of $\beta$; if $\beta =df$, they are the critical points of $f$.

A neighborhood of a compact Lagrangian submanifold $L\subset (M,\omega )$ is symplectomorphic, by a map restricting to the identity on $L$, to a neighborhood of the zero section in $(T^{\ast }L,d\lambda )$, with the sign chosen to match the convention for the canonical form.

A Lagrangian complement to $TX$ in $TM{|}_X$ is identified with $T^{\ast }X$ by $v\mapsto \omega (\cdot ,v){|}_{TX}$. This produces a vector bundle isomorphism $TX\oplus T^{\ast }X\to TM{|}_X$ that is the identity on $TX$ and carries the canonical pairing $\zeta (u)-\eta (v)$ to $\omega$. A suitably chosen tubular neighborhood map therefore matches the ambient two-forms along $X$, enabling the relative Moser theorem.

If a compact Lagrangian $L$ has $H^1(L;\mathbb{R})=0$, every sufficiently close Lagrangian graph is the graph of $df$ and meets the zero section at least at a maximum and a minimum of $f$. Without the cohomology assumption, a nowhere-zero closed one-form can give a disjoint nearby graph.

A compact Lagrangian submanifold $L$ with vanishing first de Rham cohomology cannot be a fibre of a smooth positive-dimensional fiber bundle whose nearby fibres are also Lagrangian submanifolds. In a Weinstein neighborhood, nearby fibres are one-form graphs of closed forms. They are exact by the cohomology assumption, so meet the zero section at a critical point of their potential, contradicting disjointness of distinct fibres.

If a Hamiltonian is constant on a Lagrangian $L$, then $dH$ vanishes on $TL$. Hence its Hamiltonian vector field belongs to $(TL{)}^{\omega }=TL$, and its flow preserves $L$.

A Lagrangian $L\subset M$ is displaceable by a class of isotopies when some isotopy ${\varphi }_t$ in that class satisfies ${\varphi }_1(L)\cap L=\varnothing$.

For a compact orientable Lagrangian $L$, the identification $NL\cong T^{\ast }L$ givesIf $\chi (L)\ne 0$, no smoothly isotopic copy of $L$ can be disjoint from it.