Canonical Contact Structure Research Articles

On the space of null geodesics of a spacetime: the compact case, Engel geometry and retrievability

We compute the contact manifold of null geodesics of the family of spacetimes left{ left( mathbb {S}^2times mathbb {S}^1, g_circ -frac{d^2}{c^2}dt^2right) right} _{d,cin mathbb {N}^+text { coprime}}, with g_circ the round metric on mathbb {S}^2 and t the mathbb {S}^1-coordinate. We find that these are the lens spaces L(2c, 1) together with the pushforward of the canonical contact structure on STmathbb {S}^2cong L(2,1) under the natural projection L(2,1)rightarrow L(2c,1). We extend this computation to Ztimes mathbb {S}^1 for Z a Zoll manifold. On the other hand, motivated by these examples, we show how Engel geometry can be used to describe the manifold of null geodesics of a certain class of three-dimensional spacetimes, by considering the Cartan deprolongation of their Lorentz prolongation. We characterize the three-dimensional contact manifolds that are contactomorphic to the space of null geodesics of a spacetime. The characterization consists in the existence of an overlying Engel manifold with a certain foliation and, in this case, we also retrieve the spacetime.

Some applications of Menke’s JSJ decomposition for symplectic fillings

We apply Menke’s JSJ decomposition for symplectic fillings to several families of contact 3-manifolds. Among other results, we complete the classification up to orientation-preserving diffeomorphism of strong symplectic fillings of lens spaces. We show that exact symplectic fillings of contact manifolds obtained by surgery on certain Legendrian negative cables are the result of attaching a Weinstein 2-handle to an exact filling of a lens space. For large families of contact structures on Seifert fibered spaces over S 2 S^2 , we reduce the problem of classifying exact symplectic fillings to the same problem for universally tight or canonical contact structures. Finally, virtually overtwisted circle bundles over surfaces with genus greater than one and negative twisting number are seen to have unique exact fillings.

A geometric approach to contact Hamiltonians and contact Hamilton–Jacobi theory

We propose a novel approach to contact Hamiltonian mechanics which, in contrast to the one dominating in the literature, serves also for non-trivial contact structures. In this approach Hamiltonians are no longer functions on the contact manifold M itself but sections of a line bundle over M or, equivalently, 1-homogeneous functions on a certain -principal bundle , which is equipped with a homogeneous symplectic form ω. In other words, our understanding of contact geometry is that it is not an ‘odd-dimensional cousin’ of symplectic geometry but rather a part of the latter, namely ‘homogeneous symplectic geometry’. This understanding of contact structures is much simpler than the traditional one and very effective in applications, reducing the contact Hamiltonian formalism to the standard symplectic picture. We develop in this language contact Hamiltonian mechanics in the autonomous, as well as the time-dependent case, and the corresponding Hamilton–Jacobi theory. Fundamental examples are based on canonical contact structures on the first jet bundles of sections of line bundles L, which play in contact geometry a fundamental rôle, similar to that played by cotangent bundles in symplectic geometry.

Distributions, First Integrals and Legendrian Foliations

We study germs of holomorphic distributions with “separated variables”. In codimension one, a well know example of this kind of distribution is given by $$\begin{aligned} dz=(y_1dx_1-x_1dy_1)+\dots +(y_mdx_m-x_mdy_m), \end{aligned}$$which defines the canonical contact structure on \({\mathbb {C}}{\mathbb {P}}^{2m+1}\). Another example is the Darboux distribution $$\begin{aligned} dz=x_1dy_1+\dots +x_mdy_m, \end{aligned}$$which gives the normal local form of any contact structure. Given a germ \({\mathcal {D}}\) of holomorphic distribution with separated variables in \(({\mathbb {C}}^n,0)\), we show that there exists , for some \(\kappa \in {\mathbb {Z}}_{\ge 0}\) related to the Taylor coefficients of \({\mathcal {D}}\), a holomorphic submersion $$\begin{aligned} H_{{\mathcal {D}}}:({\mathbb {C}}^n,0)\rightarrow ({\mathbb {C}}^{\kappa },0) \end{aligned}$$such that \({\mathcal {D}}\) is completely non-integrable on each level of \(H_{{\mathcal {D}}}\). Furthermore, we show that there exists a holomorphic vector field Z tangent to \({\mathcal {D}}\), such that each level of \(H_{{\mathcal {D}}}\) contains a leaf of Z that is somewhere dense in the level. In particular, the field of meromorphic first integrals of Z and that of \({\mathcal {D}}\) are the same. Between several other results, we show that the canonical contact structure on \({\mathbb {C}}{\mathbb {P}}^{2m+1}\) supports a Legendrian holomorphic foliation whose generic leaves are dense in \({\mathbb {C}}{\mathbb {P}}^{2m+1}\). So we obtain examples of injectively immersed Legendrian holomorphic open manifolds that are everywhere dense.

Surface singularities and planar contact structures

Dans cet article on démontre que si une structure de contact sur une variété de dimension trois est portée par un livre ouvert à pages planaires, alors une certaine configuration d’intersections n’apparaît dans l’homologie d’aucun de ses remplissages minimaux. On démontre de plus que les remplissages d’une telle variété de contact ne contiennent pas de surface symplectique de genre positif. En appliquant ces obstructions aux structures de contact canoniques sur les bords des singularités normales de surfaces, on montre que les bords des singularités isolées de surfaces dans l’espace complexe de dimension trois sont planaires seulement pour les singularités de type A n . En général, nous caractérisons complètement les bords planaires des singularités normales de surfaces (par leurs graphes de résolution) : ces singularités sont précisément les singularités rationnelles avec cycle fondamental réduit (aussi appelées singularités minimales). On montre aussi la non-planarité des structures de contact tendues sur certains petits L-espaces de Seifert ainsi que celle des structures de contact obtenues par la construction de Boothby–Wang appliquée aux surfaces de genre positif. De plus, on démontre que tout groupe de présentation finie est le groupe fondamental d’une fibration de Lefschetz à fibres planaires.

Heegaard Floer invariants of contact structures on links of surface singularities

Let a contact 3-manifold (Y, \xi_0) be the link of a normal surface singularity equipped with its canonical contact structure \xi_0 . We prove a special property of such contact 3-manifolds of "algebraic" origin: the Heegaard Floer invariant c^+(\xi_0)\in \mathrm {HF}^+(-Y) cannot lie in the image of the U -action on \mathrm {HF}^+(-Y) . It follows that Karakurt's "height of U -tower" invariants are always 0 for canonical contact structures on singularity links, which contrasts the fact that the height of U -tower can be arbitrary for general fillable contact structures. Our proof uses the interplay between the Heegaard Floer homology and Némethi's lattice cohomology.

Open books decompositions of links of minimally elliptic singularities

AbstractWe present an explicit Milnor open book decomposition supporting the canonical contact structure on the link of each minimally elliptic singularity whose fundamental cycle Z satisfies . For the Milnor open books whose pages have genus less than three, we give a factorization of the monodromy which does not involve any left‐handed Dehn twists around interior curves. Necessary results regarding the roots of reducible, and irreducible elements in mapping class groups are proved, and some new relations in the mapping class groups are presented.

The contact structure on the space of null geodesics of causally simple spacetimes

It is shown that the space of null geodesics of a star-shaped causally simple subset of Minkowski space is contactomorphic to the canonical contact structure in the spherical cotangent bundle of Rn. In the 3-dimensional case we prove a similar result for a large class of causally simple contractible subsets of an arbitrary globally hyperbolic spacetime applying methods from the theory of contact-convex surfaces. Moreover we prove that under certain assumptions the space of null geodesics of a causally simple spacetime embeds with smooth boundary into the space of null geodesics of a globally hyperbolic spacetime. The characteristic foliation of this boundary provides an invariant of the conformal class of the causally simple spacetime.

Generalized chain surgeries and applications

We describe the Stein handlebody diagrams of Milnor fibers of Brieskorn singularities $x^p + y^q + z^r = 0$. We also study the natural symplectic operation by exchanging two Stein fillings of the canonical contact structure on the links in the case $p = q = r$, where one of the fillings comes from the minimal resolution and the other is the Milnor fiber. We give two different interpretations of this operation, one as a symplectic sum and the other as a monodromy substitution in a Lefschetz fibration.

Genus one Lefschetz fibrations on disk cotangent bundles of surfaces

We describe a Lefschetz fibration of genus one on the disk cotangent bundle of any closed orientable surface \Sigma . As a corollary, we obtain an explicit genus one open book decomposition adapted to the canonical contact structure on the unit cotangent bundle of \Sigma .

Stein and Weinstein structures on disk cotangent bundles of surfaces

Gompf (Ann Math (2) 148(2):619–693, 1998) describes a Stein domain structure on the disk cotangent bundle of any closed surface S by a Legendrian handlebody diagram. We prove that Gompf’s Stein domain is symplectomorphic to the disk cotangent bundle equipped with its canonical symplectic structure and the boundary of this domain is contactomorphic to the unit cotangent bundle of S equipped with its canonical contact structure. As a corollary, we obtain a surgery diagram for the canonical contact structure on the unit cotangent bundle of S.

Contact structures on AR-singularity links

An isolated complex surface singularity induces a canonical contact structure on its link. In this paper, we initiate the study of the existence problem of Stein cobordisms between these contact structures depending on the properties of singularities. As a first step, we construct an explicit Stein cobordism from any contact 3-manifold to the canonical contact structure of a proper almost rational singularity introduced by Némethi. We also show that the construction cannot always work in the reverse direction: in fact, the U-filtration depth of contact Ozsváth–Szabó invariant obstructs the existence of a Stein cobordism from a proper almost rational singularity to a rational one. Along the way, we detect the contact Ozsváth–Szabó invariants of those contact structures fillable by an AR plumbing graph, generalizing an earlier work of the first author.

Magnetic curves in Sol3

Magnetic curves with respect to the canonical contact structure of the space Sol3 are investigated.

Calcul différentiel neuronal et architectures fonctionnelles

Neural Differential Calculus and Functional Architectures. The paper starts with a basic presentation of Neurogeometry as geometrizing the connectivity of primary visual areas by means of concepts of modern differential geometry. The retinotopic parameterization of orientation (OR) hypercolumns in area V1 by positions in the retina can be interpreted as a fibre bundle, namely the fibration of 1-jets of planar curves. The functional architecture defined on V1 by the cortico-cortical connections between OR hypercolumns can then be interpreted as the canonical contact structure of the 1-jets fibre bundle. The paper presents next the pinwheel structure of OR maps with their singularities. The existence of a characteristic mesh of the network of pinwheels corresponds to the fact that the OR field is a solution of the Helmholtz equation. The concepts of a phase field and of a Gaussian field can consequently be applied to the statistics of pinwheels. Lastly, the paper focuses on the relations between the pinwheels of OR maps and the fractures of direction (DR) maps. It presents a neurogeometrical model of beautiful empirical results from Nicholas Swindale. In this model, one takes into account not only preferred ORs and DRs but also the tuning curves of which they are peaks. The spatial layout of DRs with their singularities (fractures ending at pinwheels) is then modeled by a universal unfolding (in Thom's sense) of tuning curves.

New relations between G2 geometries in dimensions 5 and 7

There are two well-known parabolic split [Formula: see text] geometries in dimension 5, [Formula: see text] distributions and [Formula: see text] contact structures. Here we link these two geometries with yet another [Formula: see text] related contact structure, which lives on a [Formula: see text]-manifold. More precisely, we present a natural geometric construction that associates to a [Formula: see text] distribution a [Formula: see text]-dimensional bundle endowed with a canonical Lie contact structure. We further study the relation between the canonical normal Cartan connections associated with the two structures and we show that the Cartan holonomy of the induced Lie contact structure reduces to [Formula: see text]. This motivates the study of the curved orbit decomposition associated with a [Formula: see text] reduced Lie contact structure on a [Formula: see text]-manifold. It is shown that, provided an additional curvature condition is satisfied, in a neighborhood of each point in the open curved orbit the structure descends to a [Formula: see text] distribution on a local leaf space. The closed orbit carries an induced [Formula: see text] contact structure.

Fillings of unit cotangent bundles of nonorientable surfaces

We prove that any minimal weak symplectic filling of the canonical contact structure on the unit cotangent bundle of a nonorientable closed surface other than the real projective plane is s-cobordant rel boundary to the disk cotangent bundle of the surface. If the nonorientable surface is the Klein bottle, then we show that the minimal weak symplectic filling is unique up to homeomorphism.

Exotic Stein fillings with arbitrary fundamental group

Let G be a finitely presentable group. We provide an infinite family of homeomorphic but pairwise non-diffeomorphic, symplectic but non-complex closed 4-manifolds with fundamental group G such that each member of the family admits a Lefschetz fibration of the same genus over the two-sphere. As a corollary, we also show the existence of a contact 3-manifold which admits infinitely many homeomorphic but pairwise non-diffeomorphic Stein fillings such that the fundamental group of each filling is isomorphic to G. Moreover, we observe that the contact 3-manifold above is contactomorphic to the link of some isolated complex surface singularity equipped with its canonical contact structure.

Reeb dynamics of the link of the An singularity

The link of the $A_n$ singularity, $L_{A_n} \subset \mathbb{C}^3$ admits a natural contact structure $\xi_0$ coming from the set of complex tangencies. The canonical contact form $\alpha_0$ associated to $\xi_0$ is degenerate and thus has no isolated Reeb orbits. We show that there is a nondegenerate contact form for a contact structure equivalent to $\xi_0$ that has two isolated simple periodic Reeb orbits. We compute the Conley-Zehnder index of these simple orbits and their iterates. From these calculations we compute the positive $S^1$-equivariant symplectic homology groups for $\left(L_{A_n}, \xi_0 \right)$. In addition, we prove that $\left(L_{A_n}, \xi_0 \right)$ is contactomorphic to the Lens space $L(n+1,n)$, equipped with its canonical contact structure $\xi_{std}$.

Fillings of unit cotangent bundles

We study the topology of exact and Stein fillings of the canonical contact structure on the unit cotangent bundle of a closed surface $\Sigma_g$, where $g$ is at least 2. In particular, we prove a uniqueness theorem asserting that any Stein filling must be s-cobordant rel boundary to the disk cotangent bundle of $\Sigma_g$. For exact fillings, we show that the rational homology agrees with that of the disk cotangent bundle, and that the integral homology takes on finitely many possible values: for example, if $g-1$ is square-free, then any exact filling has the same integral homology and intersection form as $DT^*\Sigma_g$.

Symplectic fillings of lens spaces as Lefschetz fibrations

We construct a positive allowable Lefschetz fibration over the disk on any minimal (weak) symplectic filling of the canonical contact structure on a lens space. Using this construction we prove that any minimal symplectic filling of the canonical contact structure on a lens space is obtained by a sequence of rational blowdowns from the minimal resolution of the corresponding complex two-dimensional cyclic quotient singularity.