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First steps in symplectic topology
Abstract
CONTENTSIntroduction § 1. Is there such a thing as symplectic topology? § 2. Generalizations of the geometric theorem of Poincaré § 3. Hyperbolic Morse theory § 4. Intersections of Lagrangian manifolds § 5. Legendrian submanifolds of contact manifolds § 6. Lagrangian and Legendrian knots § 7. Two theorems of Givental' on Lagrangian embeddings § 8. Odd-dimensional analogues § 9. Optical Lagrangian manifoldsReferences
... Suppose that n = 1 and L = S 1 = R/2πZ is the standard circle, so that T * L ∼ = R × S 1 is a cylinder (or an open annulus). One says that a diffeomorphism A of M 2 = R × S 1 possesses the intersection property (or self-intersection property) if for any simple 3 closed curve γ ⊂ M 2 homotopic to {0} × S 1 0 ∈ R, γ ∼ = S 1 , the image of γ under the action of A intersects γ. 4 For instance, any exact symplectomorphism A of T * S 1 (e.g., any twisted rotation (p, q) → (p, q + v(p)) which is the phase flow mapping ϕ 1 of the Hamiltonian vector field v(p)∂ q with the Hamilton function p 0 v(τ ) dτ ) possesses the intersection property. Indeed, if a simple closed curve γ ⊂ T * S 1 homotopic to {0} × S 1 and the curve Aγ do not intersect then the (algebraic) area S dλ 1 of the domain S bounded by γ and Aγ is non-zero. ...
... All these results are discussed and the relevant references are given in, e.g., the works [1,4,7,8,24]. In [1,8,24], the subject is placed in the general context of the Floer homology theory which is very essential for the problems in question-according to [9], "whenever the principle of Lagrangian intersections fails (in the sense that a Lagrangian can be Hamiltonianly displaced), we obtain restrictions on the topology of the Lagrangian via computations in Floer homology". ...
... Opoȋtsev.3 That is, without self-intersections.4 One sometimes requires that Aγ ∩ γ ̸ = ∅ for any non-null-homotopic (i.e., not contractible) simple closed curve γ. ...
... Suppose that n = 1 and L = S 1 = R/2πZ is the standard circle, so that T * L ∼ = R × S 1 is a cylinder (or an open annulus). One says that a diffeomorphism A of M 2 = R × S 1 possesses the intersection property (or self-intersection property) if for any simple 3 closed curve γ ⊂ M 2 homotopic to {0} × S 1 0 ∈ R, γ ∼ = S 1 , the image of γ under the action of A intersects γ. 4 For instance, any exact symplectomorphism A of T * S 1 (e.g., any twisted rotation (p, q) → (p, q + v(p)) which is the phase flow mapping ϕ 1 of the Hamiltonian vector field v(p)∂ q with the Hamilton function p 0 v(τ ) dτ ) possesses the intersection property. Indeed, if a simple closed curve γ ⊂ T * S 1 homotopic to {0} × S 1 and the curve Aγ do not intersect then the (algebraic) area S dλ 1 of the domain S bounded by γ and Aγ is non-zero. ...
... All these results are discussed and the relevant references are given in, e.g., the works [1,4,7,8,24]. In [1,8,24], the subject is placed in the general context of the Floer homology theory which is very essential for the problems in question-according to [9], "whenever the principle of Lagrangian intersections fails (in the sense that a Lagrangian can be Hamiltonianly displaced), we obtain restrictions on the topology of the Lagrangian via computations in Floer homology". ...
... Opoȋtsev.3 That is, without self-intersections.4 One sometimes requires that Aγ ∩ γ ̸ = ∅ for any non-null-homotopic (i.e., not contractible) simple closed curve γ. ...
We present three explicit curious simple examples in the theory of dynamical systems. The first one is an example of two analytic diffeomorphisms R, S of a closed two-dimensional annulus that possess the intersection property but their composition RS does not (R being just the rotation by \pi/2). The second example is that of a non-Lagrangian n-torus L_0 in the cotangent bundle of T^n (n\geq 2) such that L_0 intersects neither its images under almost all the rotations of the cotangent bundle nor the zero section of the cotangent bundle. The third example is that of two one-parameter families of analytic reversible autonomous ordinary differential equations of the form \dot{x}=f(x,y), \dot{y}=\mu g(x,y) in the closed upper half-plane y\geq 0 such that the corresponding phase portraits for 0<\mu<1 and for \mu>1 are topologically non-equivalent. The first two examples are expounded within the general context of symplectic topology.
... The endomorphism field φ is called the Lorentz force corresponding to the magnetic field F. The Lorentz equation (1) has a Hamiltonian formulation. Let Φ be the symplectic form on the tangent bundle T M derived from the canonical two-form of the cotangent bundle T * M. Arnol'd [5,6], Anosov and Sinaȋ [4] observed that Lorentz equation is a Hamiltonian system on T M with perturbed symplectic form ...
... Example 6 The dual one-forms of the right invariant vector fields e R 1 , e R 2 , e R 3 , e R 4 are given by: ...
J-trajectories are arc length parameterized curves in almost Hermitian manifold which satisfy the equation ∇γ˙γ˙=qJγ˙. In this paper J-trajectories in the solvable Lie group Sol04 are investigated. The first and the second curvature of a non-geodesic J-trajectory in an arbitrary LCK manifold whose anti Lee field has constant length are examined. In particular, the curvatures of non-geodesic J-trajectories in Sol04 are characterized.
... Next, let c > 0 and K : S 2 → R be given. In [4], see also [5, Problems 1988[5, Problems /30, 1994[5, Problems /14, 1996, Arnol'd proposed the following question (actually in a more general setting, where S 2 is replaced by an oriented Riemannian surface ( , g)): ...
... Problem (P K ,c ), together with its generalizations, attracted the attention of many authors and has been studied via different mathematical tools, such as symplectic geometric [4,10,11,13,17] and variational arguments for multivalued functionals [6,15,19,20]. The relation between Problem (P K ,c ) and symplectic geometry can be explained as follows. ...
In this paper we adopt an alternative, analytical approach to Arnol’d problem [4] about the existence of closed and embedded K -magnetic geodesics in the round 2-sphere S 2 , where K : S 2 → R is a smooth scalar function. In particular, we use Lyapunov-Schmidt finite-dimensional reduction coupled with a local variational formulation in order to get some existence and multiplicity results bypassing the use of symplectic geometric tools such as the celebrated Viterbo’s theorem [21] and Bottkoll results [7].
... For n = 1 this is obvious (and was the motivation for Arnold's conjecture in [1]). For n > 1 it generalizes (for manifolds of nonpositive curvature) Gromov's result [31] that there is no closed exact Lagrangian submanifold in C n . ...
... Consider the space of all closed g 0 -geodesics of length ≤ c, parametrized with constant speed over the time interval [0] [1]. Since this space is compact (say, with the C 2 topology), there exists a C 2 -neighbourhood U of g in the space of metrics such that, for every g ∈ U and every closed g-geodesic q of length ≤ c, we still have ind(q) + null(q) ≤ n − 1. ...
We use a neck stretching argument for holomorphic curves to produce
symplectic disks of small area and Maslov class with boundary on Lagrangian
submanifolds of nonpositive curvature. Applications include the proof of
Audin's conjecture on the Maslov class of Lagrangian tori in linear symplectic
space, the construction of a new symplectic capacity, obstructions to
Lagrangian embeddings into uniruled symplectic manifolds, a quantitative
version of Arnold's chord conjecture, and estimates on the size of Weinstein
neighbourhoods. The main technical ingredient is transversality for the
relevant moduli spaces of punctured holomorphic curves with tangency
conditions.
... The Lorentz equation ∇γγ = qφγ has a Hamiltonian formulation. Let Φ be the symplectic form on the tangent bundle T M derived from the canonical two-form of the cotangent bundle T * M. Arnol'd (1961Arnol'd ( , 1986, Anosov and Sinaȋ (1967) observed that Lorentz equation is a Hamiltonian system on T M with perturbed symplectic form ...
J-trajectories are arc-length-parameterized curves in almost Hermitian manifolds, which satisfy the equation ∇γ˙γ˙=qJγ˙. In this paper, J-trajectories in the solvable Lie group Sol14 are investigated. J-trajectories of osculating order 2 and 3, homogeneous J-trajectories and J-trajectories in subspaceNil3 are examined.
... As is well known, geodesic equation is a Hamiltonian system on the tangent bundle of a Riemannian manifold whose Hamiltonian is the kinetic energy. Arnol'd [2,3], Anosov and Sinaȋ [1] observed that Lorentz equation is a Hamiltonian system on the tangent bundle with perturbed symplectic form. The Hamiltonian is still the kinetic energy. ...
We consider magnetic curves corresponding to the Killing magnetic fields in hyperbolic 3-space.
... To go beyond cotangent bundles, one leaves the classical variational approach and enters the theory of pseudoholomorphic curves initiated by Gromov in [27]. For closed symplectic manifolds, the non-autonomous setting is governed by the Arnold conjecture [4], giving a lower bound on the number of fixed points of Hamiltonian diffeomorphisms, and by the Conley conjecture, asserting that for a large class of symplectic manifolds every Hamiltonian diffeomorphism has infinitely many periodic points. These conjectures led to a tremendous development in the field and have been settled by now in very general forms, see, e.g., [17][18][19]32,39,40,48] for the Arnold conjecture and [23,25] for the Conley conjecture. ...
We study the relationship between a homological capacity cSH+(W) for Liouville domains W defined using positive symplectic homology and the existence of periodic orbits for Hamiltonian systems on W: if the positive symplectic homology of W is non-zero, then the capacity yields a finite upper bound to the π1-sensitive Hofer–Zehnder capacity of W relative to its skeleton and a certain class of Hamiltonian diffeomorphisms of W has infinitely many non-trivial contractible periodic points. En passant, we give an upper bound for the spectral capacity of W in terms of the homological capacity cSH(W) defined using the full symplectic homology. Applications of these statements to cotangent bundles are discussed and use a result by Abbondandolo and Mazzucchelli in the appendix, where the monotonicity of systoles of convex Riemannian two-spheres in R3 is proved.
... Then, as is well known, geodesic equation is a Hamiltonian system on T M whose Hamiltonian is the kinetic energy. Arnol'd [3,4], Anosov and Sinaȋ [2] observed that Lorentz equation is a Hamiltonian system on T M with perturbed symplectic form Ω F := Ω − π * F, where π : T M → M is the projection. The Hamiltonian is still the kinetic energy. ...
Representative examples of uniform magnetic fields are furnished by Kähler magnetic fields. From this point of view, magnetic Jacobi fields on surfaces or Kähler manifolds were investigated by Adachi and Gouda. On the contrary, Sasakian manifolds have non-uniform magnetic fields. We obtain all magnetic Jacobi fields along contact magnetic curves in 3-dimensional Sasakian space forms.
... Therefore u is regular and has curvature K(u) at each point. The K-loop problem has been largely studied since the seminal work [4] by Arnol'd. Most of the available existence results require compact target surfaces ; we limit ourselves to cite [9,12,13,[19][20][21][22][23] and references therein. ...
Given a constant k>1 and a real-valued function K on the hyperbolic plane H2, we study the problem of finding, for any ε≈0, a closed and embedded curve uε in H2 having geodesic curvature k+εK(uε) at each point.
... A very special case of intersecting submanifolds is given by the choice Q 0 = Q 1 , which corresponds to (a particular case of) the Arnold chord conjecture about the existence of a Reeb orbit starting and ending at a given Legendrian submanifold of a contact manifold, see [Arn86,Moh01], but in a possibly non-contact situation. As a trivial corollary of the theorem above we get existence results of Arnold chords for subcritical energies (cf. ...
Let (M,g) be a closed Riemannian manifold and
be a Tonelli Lagrangian. In this thesis we study the existence of orbits of the
Euler-Lagrange flow associated with L satisfying suitable boundary
conditions. We first look for orbits connecting two given closed submanifolds
of M satisfying the conormal boundary conditions: We introduce the Ma\~n\'e
critical value that is relevant for the problem and prove existence results for
supercritical and subcritical energies; we also complement these with
counterexamples, thus showing the sharpness of our results. We then move to the
problem of finding periodic orbits: We provide an existence result of periodic
orbits for non-aspherical manifolds generalizing the Lusternik-Fet Theorem, and
a multiplicity result in case the configuration space is the 2-torus.
... × I. The contact structure ξ n on T 2 × [0] [1] = R/Z × R/Z × [0, 1] = {(x, y, z)} is defined by ξ n = ker(cos(2πnz)d x−sin(2πnz)d y). A (not necessarily closed) contact 3–manifold (Y, ξ) has Giroux torsion τ (Y, ξ) ≥ n if it contains an embedded submanifold T 2 × I with the property that (T 2 × I, ξ| T 2 ×I ) is contactomorphic to (T 2 × [0, 1] ...
... In this paper by extending some latest results from hydrodynamics of incompressible ideal fluids to electrodynamics, we describe new classes of knotted/linked structures not present in the works by Ranada (and associates) and others e.g. ( Kedia et al 2013). Although these new classes of knots/links have their origins in contact geometry/topology (Kholodenko 2013), they should not be confused with the optical knots discussed by Arnol'd (1986). Surely, mathematically they can be brought into correspondence with each other. ...
Recently there had been a great deal of activity associated with various
schemes of designing both analytic and experimental methods describing knotted
structures in electrodynamics and in hydrodynamics. The majority of works in
electrodynamics were inspired by the influential paper by Ranada (1989) and its
subsequent refinements. In this work and in its companion we analyze Ranada's
results using methods of contact geometry and topology. Not only our analysis
allows us to reproduce his major results but in addition, it provides
opportunities for considerably extending the catalog of the known/obtained knot
types. In addition, it allows to reinterpret both the electric and magnetic
charges purely geometrically thus opening the possibility of treatment of
masses and charges in Yang-Mills and gravitational fields purely geometrically.
... We note that any such Lagrangian sphere in S 2 × S 2 is certainly homologous to ∆ since, by A. Weinstein's Lagrangian neighborhood theorem, all Lagrangian spheres have self-intersection number −2 . The problem of studying Lagrangian knots in symplectic manifolds was first proposed by V. I. Arnold in [1] and some results are already known in this direction , see for example the survey [8]. We observe that ∆ = {(x, x)|x ∈ S 2 } ⊂ S 2 × S 2 is a symplectic submanifold and S 2 × S 2 \ ∆ is symplectomorphic to a neighbourhood of the zero-section in T * S 2 , mapping ∆ to the zero-section. ...
The purpose of this text is to present a symplectic approach to Arnold diffusion problems, that is, the existence of orbits of perturbed integrable systems along which the action variables experience a drift whose length is independent of the size of the perturbation. We choose to focus on the construction of orbits drifting along chains of cylinders, taking for granted the existence of the chains. We however give a rather complete description of these chains, together with some elements on their symplectic features and some main ideas to prove their existence. We adopt the setting introduced by John Mather to prove the Arnold conjecture for perturbations of Tonelli Hamiltonians, which we see as the appropriate one to set out the various (and numerous) problems of the construction, and give some ideas to show how the symplectic approach may enable one to enlarge its scope.
In this paper, we settle three basic questions concerning the Gutt–Hutchings capacities from [11] which are conjecturally equal to the Ekeland–Hofer capacities from [7, 8]. Our primary result settles a version of the recognition question from [4], in the negative. We prove that the Gutt–Hutchings capacities together with the volume do not constitute a complete set of symplectic invariants for star-shaped domains with smooth boundary. In particular, we construct a smooth family of such domains, all of which have the same Gutt–Hutchings capacities and volume, but no two of which are symplectomorphic. We also establish two independence properties of the Gutt–Hutchings capacities. We prove that, even for star-shaped domains with smooth boundaries, these capacities are independent from the volume by constructing a family of star-shaped domains with smooth boundaries, whose Gutt–Hutchings capacities all agree but whose volumes all differ. We also prove that the capacities are mutually independent by constructing, for any [Formula: see text], a family of star-shaped domains, with smooth boundary and the same volume, whose capacities are all equal but the [Formula: see text]th. The constructions underlying these results are not exotic. They are convex and concave toric domains as defined in [11], where the authors also establish beautiful formulae for their capacities. A key to the progress made here is a significant simplification of these formulae under an additional symmetry assumption. This simplification allows us to identify new blind spots of the Gutt–Hutchings capacities which are used to construct the desired examples. This simplification also yields explicit computations of the Gutt–Hutchings capacities in many new examples.
We prove representation formulas for the coisotropic Hofer–Zehnder capacities of bounded convex domains with special coisotropic submanifolds and the leaf relation (introduced by Lisi and Rieser recently), study their estimates and relations with the Hofer–Zehnder capacity, give some interesting corollaries, and also obtain corresponding versions of a Brunn-Minkowski type inequality by Artstein–Avidan and Ostrover and a theorem by Evgeni Neduv.
We investigate magnetic curves in Killing submersions and we show that the bundle curvature is constant along magnetic curves with respect to the Killing vector field if and only if all vertical tubes derived from these magnetic curves are of constant mean curvature. Then we examine magnetic Jacobi fields along horizontal Killing magnetic curves in the total space of a Killing submersion. We generalize some important results obtained by O'Neill for horizontal geodesics to horizontal magnetic curves. Finally, we study magnetic Jacobi fields along horizontal Killing magnetic trajectories in 3-dimensional Sasakian space forms.
For subsets in the standard symplectic space whose closures are intersecting with coisotropic subspace we construct relative versions of the Ekeland–Hofer capacities of the subsets with respect to , establish representation formulas for such capacities of bounded convex domains intersecting with . We also prove a product formula and a fact that the value of this capacity on a hypersurface of restricted contact type containing the origin is equal to the action of a generalized leafwise chord on .
We show the smooth version of the nearby Lagrangian conjecture for the 2-dimensional pair of pants and the Hamiltonian version for the cylinder. In other words, for any closed exact Lagrangian submanifold of , there is a smooth or Hamiltonian isotopy, when M is a pair of pants or a cylinder respectively, from it to the 0-section. For the cylinder we modify a result of G. Dimitroglou Rizell for certain Lagrangian tori to show that it gives the Hamiltonian isotopy for a Lagrangian cylinder. For the pair of pants, we first study some results from pseudo-holomorphic curve theory and the planar Lagrangian in , then finally using a parameter construction to obtain a smooth isotopy for the pair of pants.
We prove the existence of immersed closed curves of constant geodesic curvature in an arbitrary Riemannian 2-sphere for almost every prescribed curvature. To achieve this, we develop a min-max scheme for a weighted length functional.
We prove the Hamiltonian unknottedness of real Lagrangian tori in the monotone , namely any real Lagrangian torus in is Hamiltonian isotopic to the Clifford torus. The proof is based on a neck-stretching argument, Gromov’s foliation theorem, and the Cieliebak–Schwingenheuer criterion.
This survey is a presentation of the five lectures on Riemannian contact geometry that the author gave at the conference “RIEMain in Contact”, 18-22 June 2018 in Cagliari, Sardinia. The author was particularly pleased to be asked to give this presentation and appreciated the organizers’ kindness in dedicating the conference to him. Georges Reeb once made the comment that the mere existence of a contact form on a manifold should in some sense “tighten up” the manifold. The statement seemed quite pertinent for a conference that brought together both geometers and topologists working on contact manifolds, whether in terms of “tight” vs. “overtwisted” or whether an associated metric should have some positive curvature. The first section will lay down the basic definitions and examples of the subject of contact metric manifolds. The second section will be a continuation of the first discussing tangent sphere bundles, contact structures on 3-dimensional Lie groups and a brief treatment of submanifolds. Section III will be devoted to the curvature of contact metric manifolds. Section IV will discuss complex contact manifolds and some older style topology. Section V treats curvature functionals and Ricci solitons. A sixth section has been added giving a discussion of the question of whether a Riemannian metric g can be an associated metric for more than one contact structure; at the conference this was an addendum to the third lecture.
Consider a smooth Hamiltonian function on the cotangent bundleof the n-dimensional torus such that its restriction on every fiber is strictly convex. Let L be a Lagrange invariant torus of the Hamiltonian flow which is homologous to the zero section. We show that, under some assumptions, L is a smooth section of the cotangent bundle.
For an immersed Lagrange submanifold W ⊂ T* X, one can define a nonnegative integer topologic invariant m(W) such that the image of H1(W; Z) under the Maslov class is equal to m(W) · Z. In this paper, the value of m(W) is calculated for the case of a two-dimensional oriented manifold X with the universal cover homeomorphic to R2 and an embedded Lagrange torus W. It is proved that if X = T2 and W is homologic to the zero section, then m(W) = 0. In all the other cases m(W) = 2. The last result is true also for a wide class of oriented properly embedded Lagrange surfaces in T*R2. The proof is based on the Gromov’s theory of pseudo-holomorphic curves. Some applications to the hamiltonian mechanics are mentioned.
We show that, up to Lagrangian isotopy, there is a unique Lagrangian torus inside each of the following uniruled symplectic four-manifolds: the symplectic vector space , the projective plane , and the monotone . The result is proven by studying pseudoholomorphic foliations while performing the splitting construction from symplectic field theory along the Lagrangian torus. A number of other related results are also shown. Notably, the nearby Lagrangian conjecture is established for , i.e. it is shown that every closed exact Lagrangian submanifold in this cotangent bundle is Hamiltonian isotopic to the zero-section.
In der Theorie dynamischer Systeme werden topologische Methoden oft dann eingesetzt, wenn die Dynamik zu kompliziert ist, um direkt Fragen wie die nach der Existenz periodischer Orbits zu beantworten.
We state the Arnold conjecture, which gives a lower bound for the number of fixed points of certain Hamiltonian diffeomorphisms. We then identify these fixed points with periodic orbits of Hamiltonian systems and with critical points of the “action functional” a function on the space of the contractible loops on the symplectic manifold, as well as the differential equation defining the flow of the gradient of this functional, called the Floer equation. This is a partial derivatives equation because it involves both the loop’s variable and that of the gradient’s flow. We begin studying the space of solutions of this equation by showing a compactness property.
In 1965, with a Comptes rendus note of Vladimir Arnold, a new discipline, symplectic topology, was born. In 1986, its (remarkable) first steps were reported by Vladimir Arnold himself. In the meantime…
Let be a contact form on a manifoldM, and L M a closed Legendrian submanifold. I prove that L intersects some characteristic for at least twice if all characteristics are closed and of the same period, and embeds nicely into the product of R2n and an exact symplectic manifold. As an application of the method of proof, the minimal action of a regular closed coisotropic submanifold of complex projective space is at most /2. This yields an obstruction to presymplectic embeddings, and in particular to Lagrangian embeddings.
From pulsar scintillations, we infer the presence of sheet-like structures in the interstellar medium; it has been suggested
that these are current sheets. Current sheets probably play an important role in heating the solar corona, and there is evidence
for their presence in the solar wind. Such magnetic discontinuities have been found in numerical simulations with particular
boundary conditions, as well as in simulations using an incompressible equation of state. Here, I investigate their formation
under more general circumstances by means of topological considerations as well as numerical simulations of the relaxation
of an arbitrary smoothly varying magnetic field. The simulations are performed with a variety of parameters and boundary conditions:
in low-, high- and of-order-unity plasma-β regimes, with periodic and fixed boundaries, with and without a friction force,
at various resolutions and with various diffusivities. Current sheets form, over a dynamical time-scale, under all conditions explored. At higher resolution they are thinner, and there is a greater number of weaker current sheets. The magnetic
field eventually relaxes into a smooth minimum energy state, the energy of which depends on the magnetic helicity, as well
as on the nature of the boundaries.
This is (mainly) a survey of recent results on the problem of the existence
of infinitely many periodic orbits for Hamiltonian diffeomorphisms and Reeb
flows. We focus on the Conley conjecture, proved for a broad class of closed
symplectic manifolds, asserting that under some natural conditions on the
manifold every Hamiltonian diffeomorphism has infinitely many (simple) periodic
orbits. We discuss in detail the established cases of the conjecture and
related results including an analog of the conjecture for Reeb flows, the cases
where the conjecture is known to fail, the question of the generic existence of
infinitely many periodic orbits, and local geometrical conditions that force
the existence of infinitely many periodic orbits. We also show how a recently
established variant of the Conley conjecture for Reeb flows can be applied to
prove the existence of infinitely many periodic orbits of a low-energy charge
in a non-vanishing magnetic field on a surface other than a sphere.
For an immersed Lagrange submanifold , one can define a nonnegative integer topologic invariant m(W) such that the image of under the Maslov class is equal to . In this paper, the value of m(W) is calculated for the case of a two-dimensional oriented manifold X with the universal cover homeomorphic to and an embedded Lagrange torus W. It is proved that if and W is homologic to the zero section, then . In all the other cases . The last result is true also for a wide class of oriented properly embedded Lagrange surfaces in . The proof is based on the Gromov's theory of pseudo-holomorphic curves. Some applications to the hamiltonian mechanics are mentioned.
What is a measurement in differential geometry? Our intuition is based on the Riemannian background where two kinds of quantities appear:
— local, like angle or curvature;
— global, like the distance between two points, where one should solve a varia- tional problem — to find the shortest path.
In 1983, soon after Charles Conley and Eduard Zehnder proved ([13], slightly extended in [7]) the first of Vladimir Arnold’s conjectures ([1], [2]) in global symplectic geometry, we discussed a very general statement about the number of intersections between two «exact» compact lagrangian submanifolds L
0 and L
1 of a symplectic manifold, isotopic through a hamiltonian isotopy. As far as I remember, Conley and I suggested looking for critical points of the action functional on the space of H
1 Spaths from L
0 to L
1, while Conley and Zehnder expressed the belief that, even though the L
2-gradient of this functional seemed to have few unbounded orbits, the Morse inequalities could be obtained by considering only the set of all bounded orbits. The three of us felt that very hard work and new ideas would be needed to carry out this program.
Consider a smooth Hamiltonian function on the cotangent bundle of the «-dimensional torus such that its restriction on every fiber is strictly convex. Let L be a Lagrange invariant torus of the Hamiltonian flow which is homologous to the zero section. We show that, under some assumptions, L is a smooth section of the cotangent bundle.
We construct the Poisson algebra associated to a singular mapping into symplectic space and show that this is an algebra of smooth functions generating solvable implicit Hamiltonian systems.
A point q in a contact manifold is called a translated point for a contactomorphism ϕ with respect to some fixed contact form if ϕ(q) and q belong to the same Reeb orbit and the contact form is preserved at q. The problem of existence of translated points has an interpretation in terms of Reeb chords between Legendrian submanifolds, and can be seen as a special case of the problem of leafwise coisotropic intersections. For a compactly supported contactomorphism ϕ of ℝ²ⁿ⁺¹ or ℝ²ⁿ × S¹ contact isotopic to the identity, existence of translated points follows immediately from Chekanov's theorem on critical points of quasi-functions and Bhupal's graph construction. In this article we prove that if ϕ is positive then there are infinitely many nontrivial geometrically distinct iterated translated points, i.e. translated points of some iteration ϕk. This result can be seen as a (partial) contact analog of the result of Viterbo on existence of infinitely many iterated fixed points for compactly supported Hamiltonian symplectomorphisms of ℝ²ⁿ, and is obtained with generating functions techniques.
In this paper, we prove the existence of infinitely many closed Reeb orbits for a certain class of contact manifolds. This result can be viewed as a contact analogue of the Hamiltonian Conley conjecture. The manifolds for which the contact Conley conjecture is established are the pre-quantization circle bundles with aspherical base. As an application, we prove that for a surface of genus at least two with a non-vanishing magnetic field, the twisted geodesic flow has infinitely many periodic orbits on every low energy level.
We show that if Q is simply connected, every exact Lagrangian cobordism
between compact, exact Lagrangians in the cotangent bundle of Q is an
h-cobordism. The result is an exercise in basic algebraic topology once one
invokes the Abouzaid-Kragh theorem.
Symplectic geometry is the language of Classical Mechanics in its
Hamiltonian formulation, and it also plays a crucial role in Quantum
Mechanics. Symplectic geometry seemed to be well understood until 1985,
when the mathematician Gromov discovered a surprising and unexpected
property of canonical transformations: the non-squeezing theorem.
Gromov's result, nicknamed the ``principle of the symplectic camel,''
seems at first sight to be an abstruse piece of pure mathematics. It
turns out that it has fundamental--and unsuspected--consequences in the
interpretations of both Classical and Quantum Mechanics, because it is
essentially a classical form of the uncertainty principle. We invite the
reader to a journey taking us from Gromov's non-squeezing theorem and
its dynamical interpretation to the quantum uncertainty principle,
opening the way to new insights.
Let ρ the density matrix of a mixed Gaussian state. Assuming that one of the Robertson–Schrödinger uncertainty inequalities is saturated by ρ, e.g. , we show that there exists a unique pure Gaussian state whose Wigner distribution is dominated by that of ρ and having the same variances and covariance ΔρX1, ΔρP1 and Δρ(X1, P1) as ρ. This property can be viewed as an analytic version of Gromov's non-squeezing theorem in the linear case, which implies that the intersection of a symplectic ball by a single plane of conjugate coordinates determines the radius of this ball. We conclude by giving a short geometric proof of the fact that pure Gaussian states are the only quantum states saturating the Robertson–Schrödinger uncertainty inequalities.
This article is devoted to V.I. Arnold, a famous mathematician who passed away in June 2010. We discuss life and times of Arnold, and review some of his seminal contributions to symplectic geometry and singularities theory which were among Arnold’s favorite subjects.
In this article, we study the existence of nontrivial solutions for a class of asymptotically linear Hamiltonian systems with Lagrangian boundary conditions by the Galerkin approximation methods and the L-index theory developed by the first author.
Contents Introduction Chapter I. The geometry of curves on §1. The elementary geometry of smooth curves and wavefronts §2. Contact manifolds, their Legendrian submanifolds and their fronts §3. Dual curves and derivative curves of fronts §4. The caustic and the derivatives of fronts Chapter II. Quaternions and the triality theorem §5. Quaternions and the standard contact structures on the sphere §6. Quaternions and contact elements of the sphere §7. The action of quaternions on the contact elements of the sphere §8. The action of right shifts on left-invariant fields §9. The duality of j-fronts and k-fronts of i-Legendrian curves Chapter III. Quaternions and curvatures §10. The spherical radii of curvature of fronts §11. Quaternions and caustics §12. The geodesic curvature of the derivative curve §13. The derivative of a small curve and the derivative of curvature of the curve Chapter IV. The characteristic chain and spherical indices of a hypersurface §14. The characteristic 2-chain §15. The indices of hypersurfaces on a sphere §16. Indices as linking coefficients §17. The indices of hypersurfaces on a sphere as intersection indices §18. Proofs of the index theorems §19. The indices of fronts of Legendrian submanifolds on an evendimensional sphere Chapter V. Exact Lagrangian curves on a sphere and their Maslov indices §20. Exact Lagrangian curves and their Legendrian lifts §21. The integral of a horizontal form as the area of the characteristic chain §22. A horizontal contact form as a Levi-Civita connection and a generalized Gauss-Bonnet formula §23. Proof of the formula for the Maslov index §24. The area-length duality §25. The parities of fronts and caustics Chapter VI. The Bennequin invariant and the spherical invariant §26. The spherical invariant §27. The topological meaning of the invariant Chapter VII. Pseudo-functions §28. The quasi-functions of Chekanov §29. From quasi-functions on the cylinder to pseudo-functions on the sphere, and conversely §30. Conjectures concerning pseudo-functions §31. Space curves and Sturm's theorem
Bibliography
We prove that there are obstructions to the existence of an exact Lagrange embedding from a closed manifold L to T * N. This may be seen as an extension of Gromov’s theorem as formulated by Lalonde and Sikorav, showing that no such embedding exists for N open. For example we answer positively a question by Lalonde and Sikorav on the non-existence of exact Lagrange embeddings from T 2 into T * S 2 . Our obstruction is in terms of the cohomology of the loop space of L and N and the map induced by the embedding in the cohomologies of these loop spaces. In particular, we give obstructions to the existence of an exact Lagrangian embedding inducing a degree-zero map from L to N. As another application of our method, we prove the Weinstein conjecture in cotangent bundles of simply connected manifolds (removing an assumption in a previous joint paper with H. Hofer).
It is shown that the 'pancake' described by Zeldovich (1970) is a general feature of the evolution of what are referred to as Lagrangian systems. It is noted that pancakes are one of several kinds of generic singularity formed at the nonlinear stage of evolution of such a system. Others include a cusp, a beak-to-beak, and a swallow tail. A complete list of singularities for the one- and two-dimensional cases is given, as is a discussion of the geometrical and certain dynamical properties of each kind of singularity. This list of singularities sets forth the elements from which the large-scale structure of the universe is constructed. The scheme presented here explains the existence of the flattened superclusters, as well as the rich clusters of galaxies by themselves and their chains.
CONTENTS
Introduction
§ 1. The Hamiltonian formalism. Simplest examples. Systems of Kirchhoff
type. Factorization of the Hamiltonian formalism for the B-phase of ³He
§ 2. The Hamiltonian formalism of systems of hydrodynamic origin
§ 3. What is Morse (LSM) theory?
§ 4. Equations of Kirchhoff type and the Dirac monopole
§ 5. Many-valued functional and an analogue of Morse theory. The periodic problem for equations of Kirchhoff type. Chiral fields in an external field
§ 6. Many-valued functions on finite-dimensional manifolds. An analogue of Morse theory
References
CONTENTS Introduction Chapter I. Symplectic geometry § 1. Symplectic manifolds § 2. Submanifolds of a symplectic space § 3. Lagrangian manifolds, bundles, maps, and singularities Chapter II. Applications of the theory of Lagrangian singularities § 4. Oscillatory integral § 5. Integral points § 6. Metamorphoses of caustics Chapter III. Contact geometry § 7. Wave fronts § 8. Singularities of fronts § 9. Metamorphoses of fronts Chapter IV. The convolution of invariants and its generalizations § 10. Vector fields tangent to a front § 11. The linearized convolution of invariants § 12. Period maps and intersection forms Chapter V. Lagrangian and Legendrian topology § 13. Lagrangian and Legendrian cobordisms § 14. Lagrangian and Legendrian characteristic classes Chapter VI. Projections § 15. Singularities of projections of surfaces to a plane § 16. Singularities of projections of complete intersections § 17. The geometry of bifurcation diagrams References Bibtex entry for this abstract Preferred format for this abstract (see Preferences) Find Similar Abstracts: Use: Authors Title Abstract Text Return: Query Results Return items starting with number Query Form Database: Astronomy Physics arXiv e-prints
The caustics of geometrical optics appear in experiments as scaled and sheared versions of the canonical forms given by catastrophe theory. However, because the longitudinal direction has a physically different status from the transverse directions, not all orientations and linear distortions are possible. Apart from the basic restriction that the caustic has to be tangential to the ray through the catastrophe point, the cuspoid catastrophes lack C − 2 degrees of freedom and the umbilics lack C − 3, where C is the co-dimension. Thus, for example, it is not possible to have a non-degenerate unsheared swallowtail caustic (C = 3) unless it is sideways on. These results hold for non-dissipative propagation in a general anisotropic and inhomogeneous medium, provided certain exceptional cases are excluded.
CONTENTS
Introduction
§ 1. The geometry of domains of possible motions
§ 2. Periodic trajectories of natural mechanical systems
§ 3. Periodic trajectories of irreversible systems
§ 4. Asymptotic solutions. Application to the theory of stability of motion
References
This paper has two aims. First, in an expository style an index theory for flows is presented, which extends the classical Morse theory for gradient flows on manifolds. Secondly, this theory is applied in the study of the forced oscillation problem of time dependent (periodic in time) and asymptotically linear Hamiltonian equations. Using the classical variational principle for periodic solutions of Hamiltonian systems a Morse theory for periodic solutions of such systems is established. In particular a winding number, similar to the Maslov index of a periodic solution is introduced, which is related to the Morse index of the corresponding critical point. This added structure is useful in the interpretation of the periodic solutions found. (A)
: Periodic solutions of Hamiltonian systems are also critical points of a function on the loop space of the underlying phase space. If this functional is bounded below, Morse's theory of critical points applies and he made such an application to the problem of closed geodesics. In the present problem (and in many more which arise in physics) the functional is not bounded below and in fact tends to + infinity and - infinity on (different) infinite dimensional sets. Understanding such 'infinitely indefinite' functionals is basic for mathematical physics. The fundamental work of P. Rabinowitz set the tone for overcoming this difficulty. It's modification here solves (the simplest version) of one of the key problems of symplectic geometry. (Author)
£*JCÜ = dHt. It was conjectured by Arnold (1), as an extension of the Poincaré-Birkhoff annulus theorem (3, 7), that every automorphism of a compact symplectic manifold P, homologous to the identity, has at least as many fixed points as a function on P has critical points. Arnold's conjecture was proven by Conley and Zehnder (4) for the torus T2n « R 2n /Z 2n with its usual symplectic structure. They show that every symplectic automorphism of T 2n , homologous to the identity, has at least n + 1 fixed points, and at least 22n if all are nondegenerate. Their method was extended in (8) to prove a version of Arnold's conjecture for arbitrary P under the additional assumption that the hamiltonian vector field £t is sufficiently C° small. In this note we announce a proof of Arnold's conjecture for the complex projective space CP n with its standard symplectic structure. We prove that a symplectic diffeomorphism of CP n, homologous to the identity, has at least n+\ distinct fixed points. (By the Lefschetz fixed point theorem, any continu ous map from CP n to itself, homotopic to the identity, has at least n +1 fixed points counted with multiplicities.) For n = 1 (CP 1 « S2) the result was al ready known (1), but with a proof which worked only in this two-dimensional case. The proof for T 2n in (4) made use of a variational principle in which the fixed points of the map were identified with periodic solutions of a time- dependent hamiltonian system and then identified with critical points of a functional on the space of contractible loops on T 2n. The corresponding functional in the case of CP n is multiple valued, and there are other difficulties connected with the curved geometry of CP n, so we need a new approach. Our trick is to consider the hamiltonian system on CP n as the reduction, in the sense of (6), of a hamiltonian system on C n+1 and then adapt recently developed methods (2) for finding periodic orbits in C n+1 . This method is similar to that of Conley and Zehnder in that a problem on a compact manifold is lifted to a problem on euclidean space invariant under a group of transformations.
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