Lagrangian submanifold
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In differential geometry, a subject of mathematics, a symplectic manifold is a
smooth manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
, M , equipped with a closed
nondegenerate In mathematics, a degenerate case is a limiting case of a class of objects which appears to be qualitatively different from (and usually simpler than) the rest of the class, and the term degeneracy is the condition of being a degenerate case. T ...
differential 2-form \omega , called the
symplectic form In mathematics, a symplectic vector space is a vector space ''V'' over a field ''F'' (for example the real numbers R) equipped with a symplectic bilinear form. A symplectic bilinear form is a mapping that is ; Bilinear: Linear in each argument ...
. The study of symplectic manifolds is called symplectic geometry or
symplectic topology Symplectic geometry is a branch of differential geometry and differential topology that studies symplectic manifolds; that is, differentiable manifolds equipped with a closed, nondegenerate 2-form. Symplectic geometry has its origins in the Ha ...
. Symplectic manifolds arise naturally in abstract formulations of
classical mechanics Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classi ...
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 Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by classi ...
; in particular, they are a generalization of the phase space of a closed system. In the same way the
Hamilton equations Hamiltonian mechanics emerged in 1833 as a reformulation of Lagrangian mechanics. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities \dot q^i used in Lagrangian mechanics with (generalized) ''momenta ...
allow one to derive the time evolution of a system from a set of
differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
s, 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''. So we require a linear map from the
tangent manifold In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M . As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points and of ...
''TM'' to the cotangent manifold ''T''''M'', or equivalently, an element of . Letting ''ω'' denote a
section Section, Sectioning or Sectioned may refer to: Arts, entertainment and media * Section (music), a complete, but not independent, musical idea * Section (typography), a subdivision, especially of a chapter, in books and documents ** Section sig ...
of , the requirement that ''ω'' be non-degenerate ensures that for every differential ''dH'' there is a unique corresponding vector field ''VH'' such that . Since one desires the Hamiltonian to be constant along flow lines, one should have , which implies that ''ω'' is alternating and hence a 2-form. Finally, one makes the requirement that ''ω'' should not change under flow lines, i.e. that the
Lie derivative In differential geometry, the Lie derivative ( ), named after Sophus Lie by Władysław Ślebodziński, evaluates the change of a tensor field (including scalar functions, vector fields and one-forms), along the flow defined by another vector fi ...
of ''ω'' along ''VH'' vanishes. Applying Cartan's formula, this amounts to (here \iota_X is the
interior product In mathematics, the interior product (also known as interior derivative, interior multiplication, inner multiplication, inner derivative, insertion operator, or inner derivation) is a degree −1 (anti)derivation on the exterior algebra of ...
): :\mathcal_(\omega) = 0\;\Leftrightarrow\;\mathrm d (\iota_ \omega) + \iota_ \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

A symplectic form on a smooth manifold M is a closed non-degenerate differential
2-form In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, ...
\omega . Here, non-degenerate means that for every point p \in M , the skew-symmetric pairing on the
tangent space In mathematics, the tangent space of a manifold generalizes to higher dimensions the notion of '' tangent planes'' to surfaces in three dimensions and ''tangent lines'' to curves in two dimensions. In the context of physics the tangent space to a ...
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 . Since in odd dimensions,
skew-symmetric matrices In mathematics, particularly in linear algebra, a skew-symmetric (or antisymmetric or antimetric) matrix is a square matrix whose transpose equals its negative. That is, it satisfies the condition In terms of the entries of the matrix, if a_ ...
are always singular, the requirement that \omega be nondegenerate implies that M has an even dimension. The closed condition means that the exterior derivative of \omega vanishes. 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.


Examples


Symplectic vector spaces

Let \ be a basis for \R^. We define our symplectic form ''ω'' on this basis as follows: :\omega(v_i, v_j) = \begin 1 & j-i =n \text 1 \leqslant i \leqslant n \\ -1 & i-j =n \text 1 \leqslant j \leqslant n \\ 0 & \text \end In this case the symplectic form reduces to a simple quadratic form. If ''In'' denotes the ''n'' × ''n'' identity matrix then the matrix, Ω, of this quadratic form is given by the block matrix: :\Omega = \begin 0 & I_n \\ -I_n & 0 \end.


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 In mathematics, the tautological one-form is a special 1-form defined on the cotangent bundle T^Q of a manifold Q. In physics, it is used to create a correspondence between the velocity of a point in a mechanical system and its momentum, thus prov ...
:\omega = \sum_^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 Solder (; NA: ) is a fusible metal alloy used to create a permanent bond between metal workpieces. Solder is melted in order to wet the parts of the joint, where it adheres to and connects the pieces after cooling. Metals or alloys suitable ...
" 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.


Kähler manifolds

A
Kähler manifold In mathematics and especially differential geometry, a Kähler manifold is a manifold with three mutually compatible structures: a complex structure, a Riemannian structure, and a symplectic structure. The concept was first studied by Jan Arn ...
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^n has a symplectic form which is the restriction of the Fubini—Study form on the projective space \mathbb^n.


Almost-complex manifolds

Riemannian manifolds In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space ''T ...
with an \omega-compatible
almost complex structure In mathematics, an almost complex manifold is a smooth manifold equipped with a smooth linear complex structure on each tangent space. Every complex manifold is an almost complex manifold, but there are almost complex manifolds that are not complex ...
are termed almost-complex manifolds. They generalize Kähler manifolds, in that they need not be
integrable In mathematics, integrability is a property of certain dynamical systems. While there are several distinct formal definitions, informally speaking, an integrable system is a dynamical system with sufficiently many conserved quantities, or first ...
. 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 In mathematics, a submanifold of a manifold ''M'' is a subset ''S'' which itself has the structure of a manifold, and for which the inclusion map satisfies certain properties. There are different types of submanifolds depending on exactly which ...
of a symplectic manifold (M, \omega) : * Symplectic submanifolds of M (potentially of any even dimension) are submanifolds S \subset M such that \omega, _S is a symplectic form on S . * Isotropic submanifolds are submanifolds where the symplectic form restricts to zero, i.e. each tangent space is an
isotropic subspace In mathematics, a quadratic form over a field (mathematics), field ''F'' is said to be isotropic if there is a non-zero vector on which the form evaluates to zero. Otherwise the quadratic form is anisotropic. More precisely, if ''q'' is a quadrati ...
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 (M,\omega) are submanifolds where the restriction of the symplectic form \omega to L\subset M is vanishing, i.e. \omega, _L=0 and \textL=\tfrac\dim M. Lagrangian submanifolds are the maximal isotropic submanifolds. In physics, Lagrangian submanifolds are frequently called
brane In string theory and related theories such as supergravity theories, a brane is a physical object that generalizes the notion of a point particle to higher dimensions. Branes are dynamical objects which can propagate through spacetime accordin ...
s. One major example is that the graph of a
symplectomorphism In mathematics, a symplectomorphism or symplectic map is an isomorphism in the category of symplectic manifolds. In classical mechanics, a symplectomorphism represents a transformation of phase space that is volume-preserving and preserves the sy ...
in the product symplectic manifold is Lagrangian. Their intersections display rigidity properties not possessed by smooth manifolds; the
Arnold conjecture The Arnold conjecture, named after mathematician Vladimir Arnold, is a mathematical conjecture in the field of symplectic geometry, a branch of differential geometry. Statement Let (M, \omega) be a compact symplectic manifold. For any smooth f ...
gives the sum of the submanifold's
Betti number In algebraic topology, the Betti numbers are used to distinguish topological spaces based on the connectivity of ''n''-dimensional simplicial complexes. For the most reasonable finite-dimensional spaces (such as compact manifolds, finite simplici ...
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 \R^_ have global coordinates labelled (x_1, \dotsc, x_n, y_1, \dotsc, y_n). Then, we can equip \R_^ with the canonical symplectic form :\omega =\mathrmx_1\wedge \mathrmy_1 + \dotsb + \mathrmx_n\wedge \mathrmy_n. There is a standard Lagrangian submanifold given by \R^n_ \to \R^_. The form \omega vanishes on \R^n_ because given any pair of tangent vectors X= f_i(\textbf) \partial_, Y=g_i(\textbf)\partial_, we have that \omega(X,Y) = 0. To elucidate, consider the case n=1. Then, X = f(x)\partial_x, Y=g(x)\partial_x, and \omega = \mathrmx\wedge \mathrmy. Notice that when we expand this out :\omega(X,Y) = \omega(f(x)\partial_x,g(x)\partial_x) = \fracf(x)g(x)(\mathrmx(\partial_x)\mathrmy(\partial_x) - \mathrmy(\partial_x)\mathrmx(\partial_x)) both terms we have a \mathrmy(\partial_x) 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 :X = \. Then, we can present T^*X as :T^*X = \ where we are treating the symbols \mathrmx,\mathrmy as coordinates of \R^4 = T^*\R^2. We can consider the subset where the coordinates \mathrmx=0 and \mathrmy=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 \mathrmf_1,\dotsc,df_k.


Example: Parametric submanifold

Consider the canonical space \R^ with coordinates (q_1,\dotsc ,q_n,p_1,\dotsc ,p_n). A parametric submanifold L of \R^ is one that is parameterized by coordinates (u_1,\dotsc,u_n) such that :q_i=q_i(u_1,\dotsc,u_n) \quad p_i=p_i(u_1,\dotsc,u_n) This manifold is a Lagrangian submanifold if the
Lagrange bracket Lagrange brackets are certain expressions closely related to Poisson brackets that were introduced by Joseph Louis Lagrange in 1808–1810 for the purposes of mathematical formulation of classical mechanics, but unlike the Poisson brackets, have fal ...
_i,u_j/math> vanishes for all i,j. That is, it is Lagrangian if : _i,u_j\sum_k \frac \frac - \frac \frac = 0 for all i,j. This can be seen by expanding : \frac = \frac \frac + \frac \frac in the condition for a Lagrangian submanifold L. This is that the symplectic form must vanish on the
tangent manifold In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M . As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points and of ...
TL; that is, it must vanish for all tangent vectors: :\omega\left( \frac , \frac \right)=0 for all i,j. Simplify the result by making use of the canonical symplectic form on \R^: : \omega\left( \frac , \frac \right) = -\omega\left( \frac , \frac \right) = 1 and all others vanishing. As 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 In mathematics, Floer homology is a tool for studying symplectic geometry and low-dimensional topology. Floer homology is a novel invariant that arises as an infinite-dimensional analogue of finite-dimensional Morse homology. Andreas Floer in ...
— this is an application of
Morse theory In mathematics, specifically in differential topology, Morse theory enables one to analyze the topology of a manifold by studying differentiable functions on that manifold. According to the basic insights of Marston Morse, a typical differentiab ...
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.


Example: Morse theory

Another useful class of Lagrangian submanifolds occur in
Morse theory In mathematics, specifically in differential topology, Morse theory enables one to analyze the topology of a manifold by studying differentiable functions on that manifold. According to the basic insights of Marston Morse, a typical differentiab ...
. Given a
Morse function In mathematics, specifically in differential topology, Morse theory enables one to analyze the topology of a manifold by studying differentiable functions on that manifold. According to the basic insights of Marston Morse, a typical differentiab ...
f:M\to\R and for a small enough \varepsilon one can construct a Lagrangian submanifold given by the vanishing locus \mathbb(\varepsilon\cdot \mathrmf) \subset T^*M. For a generic Morse function we have a Lagrangian intersection given by M \cap \mathbb(\varepsilon\cdot \mathrmf) = \text(f).


Special Lagrangian submanifolds

In the case of Kahler manifolds (or Calabi–Yau manifolds) we can make a choice \Omega=\Omega_1+\mathrm\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 hyperKahler manifolds, # fixed points of a real structure of Calabi–Yau manifolds. The
SYZ conjecture The SYZ conjecture is an attempt to understand the mirror symmetry conjecture, an issue in theoretical physics and mathematics. The original conjecture was proposed in a paper by Strominger, Yau, and Zaslow, entitled "Mirror Symmetry is ''T''- ...
deals with the study of special Lagrangian submanifolds in
mirror symmetry In mathematics, reflection symmetry, line symmetry, mirror symmetry, or mirror-image symmetry is symmetry with respect to a reflection. That is, a figure which does not change upon undergoing a reflection has reflectional symmetry. In 2D ther ...
; see . 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.


Lagrangian fibration

A Lagrangian fibration of a symplectic manifold ''M'' is a fibration where all of the
fibres Fiber or fibre (from la, fibra, links=no) is a natural or artificial substance that is significantly longer than it is wide. Fibers are often used in the manufacture of other materials. The strongest engineering materials often incorporate ...
are Lagrangian submanifolds. Since ''M'' is even-dimensional we can take local coordinates and by
Darboux's theorem Darboux's theorem is a theorem in the mathematical field of differential geometry and more specifically differential forms, partially generalizing the Frobenius integration theorem. It is a foundational result in several fields, the chief among ...
the symplectic form ''ω'' can be, at least locally, written as , where d denotes the exterior derivative and ∧ denotes the
exterior product In mathematics, specifically in topology, the interior of a subset of a topological space is the union of all subsets of that are open in . A point that is in the interior of is an interior point of . The interior of is the complement of th ...
. 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 T^*\R^n, and the Lagrangian fibration as the trivial fibration \pi: T^*\R^n \to \R^n. This is the canonical picture.


Lagrangian mapping

Let ''L'' be a Lagrangian submanifold of a symplectic manifold (''K'',ω) given by an
immersion Immersion may refer to: The arts * "Immersion", a 2012 story by Aliette de Bodard * ''Immersion'', a French comic book series by Léo Quievreux#Immersion, Léo Quievreux * Immersion (album), ''Immersion'' (album), the third album by Australian gro ...
(''i'' is called a Lagrangian immersion). Let give a Lagrangian fibration of ''K''. The composite is a Lagrangian mapping. The critical value set of ''π'' ∘ ''i'' is called a caustic. Two Lagrangian maps and are called Lagrangian equivalent if there exist
diffeomorphism In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable. Definition Given two ...
s ''σ'', ''τ'' and ''ν'' such that both sides of the diagram given on the right commute, and ''τ'' preserves the symplectic form. Symbolically: : \tau \circ i_1 = i_2 \circ \sigma, \ \nu \circ \pi_1 = \pi_2 \circ \tau, \ \tau^*\omega_2 = \omega_1 \, , where ''τ''''ω''2 denotes the
pull back In mathematics, a pullback is either of two different, but related processes: precomposition and fiber-product. Its dual is a pushforward. Precomposition Precomposition with a function probably provides the most elementary notion of pullback: i ...
of ''ω''2 by ''τ''.


Special cases and generalizations

* A symplectic manifold (M, \omega) is exact if the symplectic form \omega is exact. For example, the cotangent bundle of a smooth manifold is an exact symplectic manifold. The
canonical symplectic form In mathematics, the tautological one-form is a special 1-form defined on the cotangent bundle T^Q of a manifold Q. In physics, it is used to create a correspondence between the velocity of a point in a mechanical system and its momentum, thus prov ...
is exact. * A symplectic manifold endowed with a
metric Metric or metrical may refer to: * Metric system, an internationally adopted decimal system of measurement * An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement Mathematics In mathem ...
that is compatible with the symplectic form is an almost Kähler manifold in the sense that the tangent bundle has an
almost complex structure In mathematics, an almost complex manifold is a smooth manifold equipped with a smooth linear complex structure on each tangent space. Every complex manifold is an almost complex manifold, but there are almost complex manifolds that are not complex ...
, but this need not be
integrable In mathematics, integrability is a property of certain dynamical systems. While there are several distinct formal definitions, informally speaking, an integrable system is a dynamical system with sufficiently many conserved quantities, or first ...
. * Symplectic manifolds are special cases of a
Poisson manifold In differential geometry, a Poisson structure on a smooth manifold M is a Lie bracket \ (called a Poisson bracket in this special case) on the algebra (M) of smooth functions on M , subject to the Leibniz rule : \ = \h + g \ . Equivalent ...
. * A multisymplectic manifold of degree ''k'' is a manifold equipped with a closed nondegenerate ''k''-form. * A polysymplectic manifold is a Legendre bundle provided with a polysymplectic tangent-valued (n+2)-form; it is utilized in Hamiltonian field theory.


See also

* * − an odd-dimensional counterpart of the symplectic manifold. * * * * * * * * * *


Notes


References

* * * * * * *


External links

* * * * * * {{DEFAULTSORT:Symplectic Manifold Differential topology Symplectic geometry Hamiltonian mechanics Smooth manifolds