8. (Alejandro) Let be an arbitrary transitive homeomorphism and be an arbitrary non-constant continuous function. Then, let’s define , , and consider the suspension flow on . Note that for each and : . So the compact set is -invariant for every , and is not transitive. Notice the function c is not constant because is transitive and is not constant itself.

7. Let and be the projective action induced by . Then for , , .

So , and .

Replace by : , and ,

or: , and .

Replace be and pass : .

Let . It seems that preserves the volume on iff .

6. Let be a Riemannian manifold and be the geodesic flow.

Let be the canonical projection and be the vertical subspace.

Let , , where with and . Let .

Then , and , give the identifications and induce the following

(geodesic vector) ,

(Sasaki metric) ,

(symplectic form) ,

(almost complex structure) ,

(contact form on ) .

5. Let be an embedding and be the homotopy class of . Then is coprime.

4. (Etnyre and Ghrist pdf) Unlike foliations, contact structures are structurally stable, in the sense that not only is a perturbation of a contact form still a contact form, but also such a perturbation has kernel isotopic to that of . In fact, a standard application of the Moser method in this context implies that every contact structure is locally contactomorphic to, that is, diffeomorphic via a map which carries the contact structure, to the kernel of on .

3. Mitsumatsu’s Theorem (told by YU Bin).

Let be a 3-manifold supporting some Anosov flow. Then admits a contact structure.

Proof. Let be an Anosov flow and be the splitting. Let and be two unit vector fields, and . Then it is easy to check that is non-integrable at every . So by 1., there exists some contact form.

Since , and , we get

. So the problem is, does not preserve . In fact there exists non-transitive Anosov flow constructed by Franks and Williams (here), which can not preserve any contact form.

2. Let be a contact manifold. Then there exists a distinguished vector field , called the characterisitc vector field or Reeb field, which is determined uniquely by the following two conditions: , . The corresponding flow generated by is called the contact flow.

Proposition. The contact flow always preserves the contact form. That is, given a contact form on , we have . Equivalently, for all , .

Proof. .

On the other hand, let’s start with a flow preserving . We see that , is an -invariant smooth function. So if the flow is transitive (for example a vol-pre Anosov flow), then must be constant (up to scaling, we can assume ) and hence , which implies that .

1. Let be a closed 3 dimensional manifold, be a nonvanishing 1-form on . Then is said to be a contact form if the following holds:

is nonvanishing at every .

Let a plane-field. It is said to be a contact structure if it is a kernel of the contact form: .

The orientation induced by is independent of .

Hence has a natural orientation, either agree (a positive structure) or disagree (a negative structure) with that of .

Note: a plane-field is a contact structure iff is non-integrable at every .

Proof. (1) Let for each . Pick two local fields on spanning . Extends to a basis of . Then

So over . This shows that is non-integrable at every .

(2) Let be a plane-field and be a one form with for every such that is non-integrable at every . Let be given as above. Then

since . So is nonvanishing at every .

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According to the Frobenius integrability theorem, a contact structure is a maximally non-integrable plane-field. In particular, a contact structure is locally twisted at every point and may be thought of as an *anti-foliation*.