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Let be a differential manifold, its tangent bundle, and a field of hyperplanes on , that is, a smooth sub-bundle of codimension . Here the terms `differential' and `smooth' are used synonymously with . Locally, can be written as the kernel of a non-vanishing differential -form . A -form defined globally on with can be found if and only if is coorientable, which is equivalent to saying that the quotient line bundle is trivial. The -form is determined by up to multiplication by a smooth function or, if the coorientation of has been fixed, by a function taking positive real values only. An equation of the form , with a non-vanishing -form, is classically referred to as a Pfaffian equation.
Definition 1.1. Let be a smooth manifold of odd dimension . A contact structure on is a hyperplane field whose (locally) defining -form has the property that the -form is nowhere zero, i.e. a volume form, on its domain of definition.
Definition 1.2. A pair consisting of an odd-dimensional manifold and a contact structure on is called a contact manifold.
Definition 1.3. A -form as in Definition 1.1, defined globally on , is called a contact form on .
Occasionally the terminology strict contact manifold is used to denote a pair consisting of an odd-dimensional manifold and a contact form on it.
2.1 The standard contact structure on R2n+1
On with Cartesian coordinates
The theorem of Darboux states that locally any contact structure looks like the standard one, cf. [Geiges2008, Theorem 2.5.1].
Theorem 2.1 (Darboux).Let be a contact form on the -dimensional manifold and a point in . Then there are coordinates on a neighbourhood of such that and
Definition 2.2. Two contact manifolds , are said to be contactomorphic if there is a diffeomorphism with , where denotes the differential of . If are contact forms defining the contact structures , respectively, this is equivalent to saying that and determine the same hyperplane field, and hence equivalent to the existence of a nowhere zero function such that .
2.2 The standard contact structure on S2n+1
Let be Cartesian coordinates on . Then the standard contact structure on the unit sphere in is given by the contact form
Write for the radial coordinate on , that is, . One checks easily that for . Since is a level set of (or ), this verifies the contact condition. Alternatively, one may regard as the unit sphere in . Then the contact structure may be viewed as the hyperplane field of complex tangencies. Indeed, write for the complex structure on corresponding to the complex coordinates , that is, . Then
which means that defines at each point the -invariant subspace of . Equation (1) follows from the observation that .
Here is a further example of contactomorphic manifolds.
Proposition 2.3. For any point , the contact manifolds and are contactomorphic.
This is slightly less obvious than it may seem, since stereographic projection does not quite do the job. For a proof of this proposition, due to Erlandsson, see [Geiges2008, Proposition 2.1.8].
2.3 The space of contact elements
Let be a smooth -dimensional manifold. A contact element is a hyperplane in a tangent space to . The space of contact elements of is the collection of pairs consisting of a point and a contact element . This space of contact elements can be naturally identified with the projectivised cotangent bundle , by associating with a hyperplane the linear map , well defined up to multiplication by a non-zero scalar, with . The space is a manifold of dimension , and it carries a natural contact structure as defined in the following proposition.
Proposition 2.4. Write for the bundle projection . For , let be the hyperplane in such that is the hyperplane in defined by . Then defines a contact structure on .
Figure 2 illustrates the construction for . Here .
2.4 A non-coorientable contact structure
In the previous example, we now specialise to . Then the space of contact elements is . In terms of Cartesian coordinates on and homogeneous coordinates on , the natural contact structure on this space of contact elements is now defined globally by equation (2). For , and identifying with with coordinate , this natural contact structure can be written as
This is an example of a contact structure that is not coorientable. It lifts to a coorientable contact structure on , given by the same equation, with . Similar orientability issues arise for general . Write and for the natural contact structure on this space of contact elements. We claim the following:
- If is even, then is orientable; is neither orientable nor coorientable.
- If is odd, then is not orientable; is not coorientable, but it is orientable.
The statement about orientability of follows from the corresponding statement for . The fact that is never coorientable follows from the observation that can be identified with the canonical line bundle on (pulled back to ), which is known to be non-trivial, see [Geiges2008, Proposition 2.1.13]. In case (i), since is orientable but not coorientable, it follows that cannot be orientable. The fact that in case (ii) the contact structure is orientable is the consequence of a more general statement in the next section.
2.5 More orientability issues
Notice that a contact manifold with a coorientable contact structure is always orientable (and so is the contact structure), because a globally defined contact form gives rise to a volume form on the manifold. This gives a quicker way to see that in our previous example for odd the contact structure cannot be coorientable. But even for contact structures that need not be coorientable one has the following:
- Any contact manifold of dimension is naturally oriented.
- Any contact structure on a manifold of dimension is naturally oriented.
Statement (i) follows from the observation that the sign of the volume form does not depend on the choice of (local) -form defining the contact structure. Similarly, in case (ii) the sign of does not depend on the choice of .
2.6 Three-dimensional contact manifolds
One can easily write down examples of contact structures on some closed -manifolds. The -sphere is dealt with in Section 2.2. The contact structure from equation (3) in Section 2.4 descends to a contact structure on the -torus . On one has the contact structure . Notice that by the previous section a -dimensional contact manifold is necessarily orientable. In fact, as shown by Martinet [Martinet1971], this is the only restriction.
Theorem 2.5 [Martinet]. Every closed, orientable -manifold admits a contact structure.
2.7 Brieskorn manifolds
3 A brief history of the terminology
The concept of a contact element first appeared in systematic form in 1896 in the work of Sophus Lie [Lie1896]. His terminology was a little more specific, for instance, a contact element of the plane was called a line element (Linienelement). A contact transformation (Berührungstransformation) for Lie was defined as above, but he only considered this in the context of spaces of contact elements and their natural contact structure (which did not yet bear that name). Such contact transformations play a significant role in the work of E. Cartan, E. Goursat, H. Poincaré and others in the second half of the 19th century. For instance, the Legendre transformation in classical mechanics is a contact transformation. The study of contact manifolds in the modern sense can be traced back to the work of Georges Reeb [Reeb1952], who referred to a strict contact manifold as a `système dynamique avec invariant intégral de Monsieur Elie Cartan'. The relation with dynamical systems comes from the fact that a contact form gives rise to a vector field defined uniquely by the equations
This vector field is nowadays called the Reeb vector field of . The words `contact structure' and `contact manifold' seem to make their first appearance in the work of Boothby-Wang [Boothby&Wang1958], Gray [Gray1959] and Kobayashi [Kobayashi1959] in the late 1950s. For more historical information on contact manifolds see [Lutz1988] and [Geiges2001].
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- [Gray1959] J. W. Gray, Some global properties of contact structures, Ann. of Math. (2) 69 (1959), 421–450. MR0112161 (22 #3016) Zbl 0092.39301
- [Kobayashi1959] S. Kobayashi, Remarks on complex contact manifolds, Proc. Amer. Math. Soc. 10 (1959), 164–167. MR0111061 (22 #1925) Zbl 0090.38502
- [Lie1896] S. Lie, Geometrie der Berührungstransformationen (dargestellt von S. Lie und G. Scheffers), B. G. Teubner, Leipzig, 1896. MR0460049 (57 #45) Zbl 03630675
- [Lutz1988] R. Lutz, Quelques remarques historiques et prospectives sur la géométrie de contact, in: Conference on Differential Geometry and Topology, (Sardinia, 1988) Rend. Sem. Fac. Sci. Univ. Cagliari 58 (1988), no.suppl., 361–393. MR1122864 (93b:53025)
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