Stefan Müller

Symplectic and contact geometry and topology

Symplectic geometry has its origins in the study of classical mechanical systems such as the planetary system or the harmonic oscillator. The geodesic flow on a Riemannian manifold provides another example. The discovery of symplectic structures allows the study of such Hamiltonian dynamical systems in a coordinate-free way, and thus on manifolds (of necessarily even dimensions) other than classical phase space. By Darboux's theorem, symplectic structures have no local invariants. The study of the global features of symplectic geometry is known as symplectic topology.

Contact geometry is an odd-dimensional analog to symplectic geometry, and also has its origins in classical mechanics, such as time-dependent mechanics and Huygens' work on geometric optics. Again Darboux's theorem dictates that contact structures and contact forms have no local invariants. Modern contact topology is concerned with global and topological features of contact geometry.

The two theories are linked in many ways, either via prequantization bundles over integral symplectic manifolds, with the total space a regular contact manifold (such circle bundles are of particular interest in mechanics), or by contactization of an exact symplectic manifold, and conversely, and perhaps more prominently, via symplectization or symplectic filling of a contact manifold. Symplectic methods also enter into topological dynamics for example in dimension two, and interact with complex geometry most successfully in dimension four, and contact geometry sometimes provides insights into purely topological questions in dimension three, such as a contact geometric proof of Cerf’s theorem, or information about knots in three-manifolds.

A symplectic structure (or also, a symplectic form) ω is a closed and non-degenerate two-form, and thus provides a means of identifying smooth vector fields with differential one-forms. A smooth vector field in turn can be integrated to define an isotopy of the underlying manifold, and this correspondence is again one-to-one, by the classical theory of ordinary differential equations. Smooth isotopies preserving the symplectic structure correspond to closed one-forms under this identification. The smooth isotopies corresponding to exact one-forms are called Hamiltonian. An exact one-form is by definition the differential of a smooth function on the underlying symplectic manifold, and assuming a suitable normalization of smooth functions, this correspondence can also be made unique. In short, every smooth function gives rise to a unique Hamiltonian isotopy, and conversely, the generating Hamiltonian function of such a Hamiltonian isotopy is unique (up to normalization). The same remarks apply to time-dependent Hamiltonian functions. An important consequence of this one-one correspondence is that invariants of Hamiltonian isotopies (and sometimes of their time-one maps) can be defined via (invariants of) their generating Hamiltonian functions. The automorphisms of a symplectic structure are symplectic diffeomorphisms (the diffeomorphism preserving the symplectic structure). A Hamiltonian dynamical system transforms under such a symplectic change of coordinates in the expected manner.

A contact structure ξ is a coorientable (or transversely orientable) nowhere integrable (or maximally non-integrable) field of hyperplanes in the tangent bundle of the contact manifold; it can be described globally as the kernel of a contact form, that is, a one-form α such that the top dimensional differential form α ∧ (dα)^n never vanishes. In contrast to symplectic isotopies, every isotopy that preserves the contact structure is Hamiltonian; contracting the contact form with the smooth vector field obtained from differentiating the contact isotopy defines a smooth function on the underlying manifold. In turn, every smooth function (the generating Hamiltonian function) defines a unique contact vector field that integrates to an isotopy that preserves the contact structure (but not necessarily the contact form). In short, a smooth function together with a contact form determine a unique contact isotopy, and vice versa. Again the same discussion applies to time-dependent Hamiltonian functions, and invariants of contact isotopies coincide with invariants of their generating Hamiltonian functions. A transformation law as in the symplectic case holds for contact dynamical systems and contact diffeomorphisms (the diffeomorphisms preserving the contact structure, or conformally rescaling the contact form).

It follows in particular that every smooth function on the underlying manifold gives rise to an automorphism of the symplectic or contact structure, and thus the groups of symplectic and contact diffeomorphisms are quite large (infinite-dimensional Lie groups). In fact, by A. Banyaga's contributions to Klein's Erlanger program, a symplectic or contact structure is determined completely by its automorphism group (up to conformal rescaling by a non-zero constant in the former case). The study of these automorphism groups has a long and fruitful history.

Topological phenomena and topological dynamics

Returning to the beginning, it is natural from a physics point of view to consider dynamical systems (and transformations) with low order of regularity. But a priori, symplectic and contact geometry are smooth theories. For example, a symplectic or contact diffeomorphism must be at least continuously differentiable for its very definition to make sense. Moreover, in both the case of a symplectic or a contact structure, if a smooth function is continuously differentiable, and its differential is at least (locally) Lipschitz continuous, then it determines a unique isotopy as explained in the preceding paragraphs. However, when the regularity of the Hamiltonian function is less than differentiable with (locally) Lipschitz derivative, the methods of ordinary differential equations are no longer available. This is the situation in which topological Hamiltonian dynamics and topological contact dynamics come into play.

If a sequence of symplectic diffeomorphisms uniformly converges to another diffeomorphism, then the limit is again symplectic (Eliashberg-Gromov). Other striking phenomena in symplectic and contact topology and dynamics that are topological in nature include the non-degeneracy of Hofer’s metric (Hofer, Lalonde-McDuff), and related to this the (existence and) behavior of symplectic capacities (Ekeland-Hofer, and many others). Moreover, the important action selectors (or spectral invariants) in Hamiltonian Floer theory depend continuously on the generating Hamiltonian (and continuously on the given quantum (co-)homology class). Yet another fascinating example is that the property of being a Hamiltonian loop (up to homotopy in the group of all diffeomorphisms) is preserved under small perturbations of the symplectic structure (Lalonde-McDuff-Polterovich), or the fact that the corresponding Seidel element depends only on the homotopy class (again in the group of all diffeomorphisms) of a Hamiltonian loop (Banyaga-Saunders). Due to the close relationship between the symplectic and contact worlds, the above remarks apply almost verbatim to contact structures as well. These topological phenomena, as well as more recent developments, have revealed the need for a generalization of Hamiltonian and contact dynamical systems to topological dynamical systems with non-smooth Hamiltonian functions and transformations. The study of various C^0-phenomena in symplectic topology, such as the C^0-robustness of the Poisson bracket induced by a symplectic form, has attracted much recent attention by many mathematicians.

Topological Hamiltonian and contact dynamics

Topological Hamiltonian dynamics and topological contact dynamics are natural and genuine extensions of smooth Hamiltonian and contact dynamics to topological dynamics. A topological Hamiltonian or contact dynamical system consists of a topological Hamiltonian or contact isotopy (a continuous isotopy of homeomorphisms), together with a possibly non-smooth topological Hamiltonian function on the underlying symplectic or contact manifold, and a topological conformal factor (a continuous function) in the contact case. By definition, this Hamiltonian function is the limit of a sequence of smooth (time-dependent) Hamiltonian functions with respect to the usual Hofer metric. Moreover, the corresponding sequence of smooth Hamiltonian or contact isotopies converges uniformly to the above continuous isotopy, and their smooth conformal factors converge uniformly to the above continuous function associated to the limit isotopy. In other words, we consider limits of smooth Hamiltonian and contact dynamical systems with respect to a metric that combines the topological C^0-metric with the dynamical Hofer metric. Similarly, topological automorphisms of a symplectic or contact structure or form are C^0-limits of symplectic or contact diffeomorphisms (together with their conformal factors in the contact case).

The objects of these theories are non-smooth in general. Nevertheless, ample evidence asserts they comprise the correct topological analogs of Hamiltonian and contact dynamics. For instance, a topological Hamiltonian or contact isotopy is not generated by a vector field, and may not even be Lipschitz continuous; nonetheless, it is uniquely determined by its associated topological Hamiltonian function, and in turn, in the contact case it determines uniquely its topological conformal factor. Composition and inversion can therefore be defined as in the smooth case, and the usual transformation law continues to hold. Conversely, every topological Hamiltonian or contact isotopy possesses a unique topological Hamiltonian function. The topological automorphism groups of a contact structure and a contact form exhibit surprising rigidity properties analogous to the well-known Eliashberg-Gromov rigidity in the case of a symplectic structure. The uniqueness theorems applied to smooth Hamiltonian and contact dynamical systems prove rigidity of Hamiltonian and contact isotopies (and their conformal factors) in the following sense. If a sequence of Hamiltonian or contact isotopies (and their conformal factors) are uniformly Cauchy, and the generating Hamiltonian functions converge (with respect to the Hofer metric) to another (continuously differentiable) Hamiltonian function (with uniquely integrable Hamiltonian or contact vector field), then the limit of the Hamiltonian or contact isotopies coincides with the Hamiltonian or contact isotopy generated by the limit Hamiltonian function (and likewise for the conformal factors). A converse rigidity result also holds for Cauchy sequences of Hamiltonian functions.

The uniqueness theorems for topological Hamiltonian and contact dynamical systems, rigidity, the group structures, and the transformation law, provide ample evidence that Hamiltonian and contact dynamics are in fact topological theories, and a priori smooth invariants can be extended to topological Hamiltonian and contact dynamical systems. In addition, these topological theories have applications to smooth Hamiltonian and contact dynamics as well as to topological dynamics. For example, on every symplectic or contact manifold there exist pairs of smooth Hamiltonian or contact vector fields that are topologically conjugate but are not conjugate by symplectic or contact C^1-diffeomorphisms. Moreover, one can answer important questions of V. I Arnold regarding the helicity of a volume preserving isotopy, and its behavior under conjugation by volume preserving homeomorphisms, provided the isotopy and the transformation can be described as lifts from a surface.

The relation between topological Hamiltonian and contact dynamics is the same as in the smooth case. More precisely, topological contact dynamics is the odd-dimensional analog of topological Hamiltonian dynamics, and a natural extension of topological strictly contact dynamics. Topological Hamiltonian dynamics of an integral symplectic manifold is intimately related to topological strictly contact dynamics of the total space of the associated Boothby-Wang prequantization bundle (Banyaga-Spaeth), and in turn, topological contact dynamics of a contact manifold corresponds to (admissible) topological Hamiltonian dynamics of its symplectizations. One can also consider a simultaneous extension of the dynamics of smooth symplectic vector fields and of topological Hamiltonian dynamics to topological symplectic dynamics (Banyaga), and topological Hamiltonian dynamics on other types of noncompact symplectic manifolds that appear for example in the context of symplectic field theories (e.g. symplectic manifolds with cylindrical ends). In another direction, one can replace the Hofer metric in the constructions outlined above by other suitable metrics, for example Viterbo-type distances, or study (abstract) completions with respect to the Hofer or Viterbo metric alone.

Every orientable surface or three-manifold admits a symplectic or contact structure, respectively, and so far many of the applications (such as the study of area-preserving homeomorphism groups and the helicity mentioned above) of topological Hamiltonian and contact dynamics to topological dynamics occur in these low dimensions.

More

Refer to my annotated publications and preprints list for more detailed information on specific results. The books "Introduction to symplectic topology" by D. McDuff and D. Salamon, V. I. Arnold's "Mathematical methods of classical mechanics," and "An introduction to contact topology" by H. Geiges are good starting points for those interested in learning more about the smooth theories.