Contact geometry

The mathematical field of contact geometry is a sub-field of differential geometry that deals with certain geometric structures on differentiable manifolds , namely with completely non-integrable fields of hyperplanes in the tangential bundle , so-called contact structures . The geometrical idea described in this way is quite simple: For each point of the manifold a plane is selected, whereby an additional condition excludes the special case that the planes lie in layers, as they are shown in the second picture.

Contact geometry has its origin, among other things, in geometric optics and thermodynamics . At the end of the 19th century, the Norwegian mathematician Sophus Lie described so-called contact transformations in detail, among other things to study differential equations and methods such as the Legendre transformation and the canonical transformation of classical mechanics . Touch transformations gave the area its name; In today's language they are images that contain contact structures and are called contactomorphisms.

Today, contact structures are studied because of their diverse topological properties and their numerous connections with other areas of mathematics and physics, such as symplectic and complex geometry , the theory of the scrolling of codimension , dynamic systems and knot theory . The relationship to symplectic geometry is particularly close, because in many ways contact structures that exist in odd dimensions are counterparts to symplectic structures in even dimensions. ${\displaystyle 1}$

A level field that is a contact structure ${\displaystyle \mathbb {R} ^{3}}$

Geometric idea

A level field on which a foliation dates ${\displaystyle \mathbb {R} ^{3}}$

The pictures on the right show excerpts from plane distributions , which means that a plane is selected for each point in three-dimensional space , from which a small piece has been drawn. In the picture below, the selected planes fit together in such a way that they can be interpreted as tangent planes to a foliation; such distributions are called integrable. In the picture above, however, this is not the case, and not even in a single point. A level distribution with this property is called completely non-integrable or at most non-integrable and defines a contact structure. ${\displaystyle \mathbb {R} ^{3}}$

In fact, every contact structure looks a little bit like the one shown in the picture; this is the statement of Darboux's theorem below . Contact geometry therefore raises the question of how contact structures on different manifolds can look on a large scale and what properties they have.

Definitions

A contact structure on a manifold is given by a field of hyperplanes , i.e. for each point of the specification of a hyperplane in the tangent space , whereby this field must be completely non-integrable. The Frobenius theorem states when a plane distribution as can be integrated; a contact structure is therefore a level distribution that does not meet the conditions of this theorem in any point. ${\displaystyle M}$ ${\displaystyle \xi }$ ${\displaystyle p}$ ${\displaystyle M}$ ${\displaystyle T_{p}M}$ ${\displaystyle \xi }$

It is often more practical not to work directly with contact structures, but with contact forms. A contact form is a differentiable 1-form on a (2n + 1) -dimensional manifold that is nowhere degenerate in the following sense: ${\displaystyle \alpha }$ ${\displaystyle M}$

${\displaystyle (\alpha \wedge ({\text{d}}\alpha )^{n})_{p}\neq 0\,}$ for all ${\displaystyle p\in M,}$

in which

${\displaystyle ({\text{d}}\alpha )^{n}=\underbrace {{\text{d}}\alpha \wedge \dots \wedge {\text{d}}\alpha } _{n},}$

that is, that is a volume shape . ${\displaystyle \alpha \wedge ({\text{d}}\alpha )^{n}}$

A contact form clearly defines a contact structure. The core of is a hyperplane field and the above condition has the effect, according to Frobenius' theorem, that there is a contact structure: This field is completely non-integrable. ${\displaystyle \xi }$ ${\displaystyle \alpha }$ ${\displaystyle \xi }$

Conversely, also according to Frobenius' theorem, an integrable plane field can be described locally as the core of a closed (a differential form is called closed if ) 1-form; on the other hand, a contact structure is a plane field that can be described locally as the core of a contact form. ${\displaystyle \alpha }$ ${\displaystyle {\text{d}}\alpha =0}$

Equivalent to the condition above, one can demand the non-degeneracy of by stating that a symplectic form should be on the hyperplane field. ${\displaystyle \alpha }$ ${\displaystyle {\text{d}}\alpha }$

Naming conventions

A manifold with a contact structure is called a contact manifold . A diffeomorphism between contact manifolds that is compatible with the respective structures is called a contactomorphism, originally in Lie contact transformation. The two contact manifolds are then called contactomorphic. ${\displaystyle M}$ ${\displaystyle \xi }$ ${\displaystyle (M,\xi )}$

Examples

A contact structure on : the level field ${\displaystyle \mathbb {R} ^{3}}$

Dimension 1

In dimension 1, the contact shapes exactly match the volume shapes, and the contact structures are accordingly trivial.

Dimension 3

A basic example of a contact structure is , equipped with the coordinates and the 1-form ${\displaystyle \mathbb {R} ^{3}}$ ${\displaystyle (x,y,z)}$

${\displaystyle {\text{d}}z-y{\text{d}}x.}$

The contact plane at a point is defined by the vectors ${\displaystyle \xi }$ ${\displaystyle (x,y,z)}$

${\displaystyle X_{1}=\partial y}$

and

${\displaystyle X_{2}=\partial x+y\partial z}$

stretched. This contact structure is shown at the top of the article.

In the picture on the right is the contact structure

${\displaystyle {\text{d}}z-r^{2}{\text{d}}\theta }$

where cylindrical coordinates are for . ${\displaystyle (r,\theta ,z)}$ ${\displaystyle \mathbb {R} ^{3}}$

Dimension 2 n + 1

The basic example on can be generalized to the standard contact shape on : On , equipped with the coordinates , define the two shapes ${\displaystyle \mathbb {R} ^{3}}$ ${\displaystyle \mathbb {R} ^{2n+1}}$ ${\displaystyle \mathbb {R} ^{2n+1}}$ ${\displaystyle (x,y,z)=(x_{1},\ldots ,x_{n},y_{1},\ldots ,y_{n},z)}$ ${\displaystyle 1}$

${\displaystyle \alpha ={\rm {d}}z-\sum _{j=1}^{n}y_{j}\,{\rm {d}}x_{j}}$

and

${\displaystyle \alpha ={\rm {d}}z+\sum _{j=1}^{n}(x_{j}\,{\rm {d}}y_{j}-y_{j}\,{\rm {d}}x_{j})}$

two mutually contactomorphic contact structures.

Reeb vector fields

A contact form contains more information than a contact structure, among other things it also defines a Reeb vector field . This vector field is defined for each point by the unique vector , so that and . ${\displaystyle R}$ ${\displaystyle {\rm {d}}\alpha (R,\cdot )=\,0}$ ${\displaystyle \alpha (R)=1}$

The Weinstein Conjecture states that every Reeb vector field has closed orbites.

Contact geometry and symplectic geometry

Let be a symplectic manifold with a convex edge, in a neighborhood of . Then a contact structure is defined . It is said that the contact manifold is symplectic fillable (ger .: symplectically fillable ). ${\displaystyle (W,\omega )}$ ${\displaystyle \omega =d\lambda }$ ${\displaystyle \partial W}$ ${\displaystyle \lambda }$ ${\displaystyle \partial W}$ ${\displaystyle (\partial W,\lambda )}$

A Liouville area is a symplectic filling in which with is completely defined. ${\displaystyle \lambda }$ ${\displaystyle \omega =d\lambda }$ ${\displaystyle W}$

A weak symplectic fill of a contact manifold is a symplectic manifold with and conformally equivalent to . In dimensions , every weakly fillable contact structure can also be filled symplectically, which is why one considers a weaker concept of weak fillability in these dimensions: is a weak filling of if is for everyone . ${\displaystyle (M,\xi =\ker(\lambda ))}$ ${\displaystyle (W,\omega )}$ ${\displaystyle M=\partial W}$ ${\displaystyle \omega _{\xi }:=\omega \mid _{\xi }}$ ${\displaystyle d\lambda \mid _{\xi }}$ ${\displaystyle \geq 5}$ ${\displaystyle (W,\omega )}$ ${\displaystyle (M,\lambda )}$ ${\displaystyle \lambda \wedge (w_{\xi }+td\lambda )^{n-1}>0}$ ${\displaystyle t\geq 0}$

Legendre submanifolds

The most interesting submanifolds of a contact manifold are its Legendre submanifolds . The non-integrability of the contact hyperplane field on a -dimensional manifold means that no -dimensional submanifold has this field as a tangent bundle , not even locally. However, -dimensional ( embedded or immersed ) submanifolds can be found whose tangent spaces lie in the contact field, i.e. H. ${\displaystyle (M,\xi )}$ ${\displaystyle (2n+1)}$ ${\displaystyle 2n}$ ${\displaystyle n}$ ${\displaystyle L}$

${\displaystyle {\text{T}}_{p}L\subset \xi }$for everyone . ${\displaystyle p\in M}$

One can show that such, so-called isotropic , submanifolds reach at most dimension . A submanifold of this maximum dimension is called a Legendre submanifold. Two such submanifolds are called equivalent if they can be connected by a family of contactomorphisms of the surrounding manifold, based on the identity. ${\displaystyle n}$

If one considers contact manifolds of dimension three, then the connected compact Legendre manifolds form nodes , which are called Legendre nodes . Each node with its standard contact structure (see the example above) can be implemented as a Legendre node. Sometimes this can be done in several ways: Inequivalent Legendre nodes can be equivalent as ordinary nodes. ${\displaystyle \mathbb {R} ^{3}}$

From symplectic field theory , invariants of Legendre submanifolds, called relative contact homology , can be obtained, which can sometimes differentiate between different Legendre submanifolds which are topologically identical.

Darboux's theorem

→ Main article : Darboux's theorem

Darboux's theorem is a fundamental result of contact geometry as well as the better known symplectic geometry . It is named after Jean Gaston Darboux , who thus solved the Pfaff problem (after Johann Friedrich Pfaff ). One of the consequences of the theorem is that any two contact manifolds of the same dimension are always locally contactomorphic. So there are for each point of an environment that includes, and local coordinates on where the contact structure has the above mentioned standard form. ${\displaystyle p}$ ${\displaystyle M}$ ${\displaystyle U}$ ${\displaystyle p}$ ${\displaystyle U}$

Compared to Riemannian geometry , this is a strong contrast: Riemannian manifolds have interesting local properties such as curvature . In contrast, on the small scale, there is only one single model of contact manifolds. For example, the two contact structures described and illustrated are on locally and can therefore kontaktomorph (grain of salt) are both standard contact structure to be mentioned. ${\displaystyle \mathbb {R} ^{3}}$ ${\displaystyle \mathbb {R} ^{3}}$

3-dimensional contact geometry

"Tight" vs. "overtwisted"

In the three-dimensional contact geometry there is the dichotomy between strict (germ .: tight ) and overwrought (ger .: overtwisted ) contact structures. A contact structure is said to be over-turned if there is an "over-turned" circular disk (i.e. the induced foliation of which has a singularity inside and otherwise orbits running from the singularity into the edge of the circular disk). The contact structure is called tight if there is no overturned circular disc. ${\displaystyle (M,\xi )}$

If a contact structure can be (weakly) symplectically filled, then it is tight. The inclusions

${\displaystyle \left\{\mathrm {symplektisch\ f{\ddot {u}}llbar} \right\}\subset \left\{\mathrm {schwach\ symplektisch\ f{\ddot {u}}llbar} \right\}\subset \left\{{\text{straff}}\right\}}$

are real inclusions in dimension 3 . In addition to overturned circular disks, the non - disappearance of the Giroux torsion is another obstacle to symplectic fillability. The Giroux torsion is defined as the maximum for which an embedding of in exists. ${\displaystyle n}$ ${\displaystyle (T^{2}\times I,\ker(\cos(2\pi nt)dx+\sin(2\pi nt)dy))}$ ${\displaystyle (M,\xi )}$

An h-principle applies to over-twisted contact structures , their classification up to homotopy is equivalent to the classification of the plane fields up to homotopy.

Open books

Giroux's theorem: On orientable 3-manifolds there is a bijection between the set of contact structures except for isotopic and the set of open books except for positive stabilization.

Boothby-Wang's theorem

A contact manifold is called regular if every point has a neighborhood through which every integral curve of the Reeb vector field passes at most once.

Boothby-Wang's theorem characterizes compact regular contact manifolds : these are precisely the -bundles over symplectic manifolds whose symplectic form determines an integral cohomology class . ${\displaystyle (M,\eta )}$ ${\displaystyle S^{1}}$ ${\displaystyle \pi \colon M\to N}$ ${\displaystyle N}$ ${\displaystyle \omega }$ ${\displaystyle [\omega ]\in H^{2}(N;\mathbb {Z} )}$

Furthermore, there is a zero-zero function in this case , so that the Reeb vector field produces the effect, and it holds . ${\displaystyle \tau \colon M\to \mathbb {R} }$ ${\displaystyle \eta ^{\prime }:=\tau \eta }$ ${\displaystyle S^{1}}$ ${\displaystyle d\eta ^{\prime }=\pi ^{*}\omega }$

literature

Introductory texts on contact geometry

• Emmanuel Giroux : Topologie de contact en dimension 3 . In: Séminaire Bourbaki . No. 760 (1992/93) , pp. 7-33 ( online ).
• John Etnyre: Introductory lectures on contact geometry . In: Proceedings of Symposia in Pure Mathematics . No. 71 , 2003, p. 81-107 , arxiv : math.SG/0111118 .
• Hansjörg Geiges : Contact geometry . In: Franki Dillen and Leopold Verstraelen (Eds.): Handbook of Differential Geometry . tape 2 . North-Holland, Amsterdam 2006, p. 315-382 , arxiv : math.SG/0307242 .
• Hansjörg Geiges Christiaan Huygens and Contact Geometry , University of Cologne, June 2005, PDF file (accessed April 8, 2014)

history

• Sophus Lie , Georg Scheffers : Geometry of touch transformations . Chelsea Publishing, Bronx NY 1977, ISBN 0-8284-0291-4 ( limited preview in the Google book search - first edition: Teubner, Leipzig 1896).
• Robert Lutz: Quelques remarques historiques et prospectives sur la géométrie de contact. In: Gruppo Nazionale di Geometria delle Varietà Differentialiabili (Ed.): Proceedings of the conference on differential geometry and topology. (Cala Gonone / Sardinia). 1988, 09. 26-30 (= Rendiconti del Seminario della Facoltà di Scienze della Università di Cagliari. 58, Suppl., ISSN  0370-727X ) Tecnoprint, Bologna, 1988, pp. 361-393.
• Hansjörg Geiges: A brief history of contact geometry and topology . In: Expositiones Mathematicae . No. 19 , 2001, ISSN  0723-0869 , p. 25-53 .

Web links

• Manifold Atlas: Contact Manifold

Individual evidence

1. ^ Sophus Lie , Georg Scheffers : Geometry of contact transformations . BJ Teubner, 1896 ( limited preview in Google book search).
2. ^ Massot-Niederkrüger-Wendl: Weak and strong fillability in higher dimensions. Inventiones Mathematicae 2013