Lagrangian submanifold

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.

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.

A simple closed curve dividing a symplectic two-sphere into two regions of equal area cannot be displaced by a symplectic isotopy. Any disjoint image would lie in one complementary disc, yet it would still have to bound a disc of half the total area.