Reeb Fields and Contact Fields.

In summary: For a Reeb field, Rw preserves not just the contact structure, but the contact form itself. So, every Reeb field is a contact field, but not otherwise. *In the case of the Reeb field Rw, its flow \Psi is a sort of path of contact-form-preserving maps, i.e., for each teach \Psit preserves w.
  • #1
WWGD
Science Advisor
Gold Member
6,910
10,290
Hi again:

I'm curious as to someone understands well the difference between a Reeb Vector Field and a general Contact field; a Contact field is one whose flow (even when only defined locally, i.e., in non-complete manifolds) preserves the contact structure, but not the form, i.e., if C is a Contact field with local flow[itex]\Phi[/itex] , and w is the contact form, then:

[itex]\Phi[/itex]*w=gw , where g is a smooth nowhere-zero function ; here gw and w are equivalent, in that the kernel of gw is the same as that of w . Similarly, we have that [itex]\Phi[/itex]* takes contact planes to contact planes, i.e., the basis tangent vectors for the contact plane at p are pushed forward to the tangent plane at (p+t).

OTOH, we have, for a Reeb field Rw for w, that Rw preserves not just the contact structure, but the contact form itself, i.e., if the flow of Rw is given by [itex]\Psi[/itex], then [itex]\Psi[/itex]*(w)=w .

So every Reeb field is a contact field, but not otherwise. I guess in the case of the Reeb field Rw, its flow [itex]\Psi[/itex] is a sort of path of contact-form-preserving maps, i.e., for each teach [itex]\Psi[/itex]t preserves w. Since Rw is also a contact field, I guess in the case of the Reeb field we can somehow normalize the function g (since the flow a Contact field C takes w to gw ), so that g==1.

Does anyone understand well what is going on here?
 
Physics news on Phys.org
  • #2
Everything you've written down makes sense. What's your actual question? Do you understand flows and pushforwards?
 
  • #3
Ben Niehoff said:
Everything you've written down makes sense. What's your actual question? Do you understand flows and pushforwards?

Yes, to a pretty good extent, I think. I would just like to know in some sense how/why a Reeb field is "stronger" than a non-Reeb contact field. Still, let me think things thru again see if I can clarify my question some more. Thanks, Ben.
 
  • #4
Actually, this is a more specific question: there is a result that a contact field Vthat is transverse to the contact pages is a Reeb field. I'm trying to show this is so; from other results, it follows that if the field V is transverse, then it will preserve not just the contact structure, but the contact form itself. I'm trying to show this.

Please let me ramble-on a bit, see if my overall knowledge of Contact Structures is accurate; I will be giving a talk a few months from now and I want to see if/where I have gaps(more likely where than if ;) ), and I'm trying to put all these terms together. Please give me some time until I learn this version of Latex.

Re my understanding, it is mostly formal, but not very geometric (given this is low-dimensional topology/geometry).

I understand the flow associated to a vector field V in a manifold M to be a curve C(t); C:(-e,e)-->M with

C'(t)=V(C(t)), i.e., the derivative of C at t coincides with the value of the vector field at that point,

and that flows are guaranteed to exist (at least) locally by , I think, one of Picard's theorems.

More rigorously, the flow ψ of V maps a point (t,p) in ℝxM-->M by sending (p,t) to the curve C(t)

with:

C(0)=p ; C'(t)=V(C(t))

The pushforward ψ* associated with a map, say the flow , maps tangent vectors at

TpM to TpM , mapping a deriva

I'm also having trouble understanding or seeing a more geometric interpretation of the result
LRw ω=0 , where L is the Lie derivative of ω. This means that the form ω is constant along the flow, but, what does this mean? I have some idea of what a constant vector field is, but I'm having trouble digesting what a constant tensor field is, or even what dω(Rw,.)=0 means geometrically.

So, to be more specific in my questions:

i)How do we show that if V is a contact field for ω , and V is transverse to the contact planes, then V is a Reeb field for ω .

ii)How does one interpret the result L ω =0 ? This says, AFAIK, that ω is constant along the flow. BUT: what is the meaning of a form being constant, or a tensor field being constant?

Thanks.
 
  • #5
Sorry if I rambled-on too much. My basic question is this:

How to we show that a contact vector field (i.e., a v.field whose flow preserves the contact structure--but not necessarily the contact form, other than up to a multiple of the form by a nonzero smooth function) that is transverse to the contact planes is a Reeb field?

A contact field is one whose flow preserves the contact form, meaning , for w the contact form , $$V_*$$ the flow of V, we have $$ V_*(w)=gw$$ , but the flow $$R_w* $$ of a Reeb field $$ R_w$$ preserves the form itself, i.e., $$R_w (w)* =w$$ , and the Reeb field satisfies:

$$ w(R_w) $$ =1 , which basically says $$w(R_w) $$ is never 0.

Now, I can see why $$ w(R_w)$$ is not zero if $$ R_w$$ is transverse, since this means $$R_w$$ is never in the contact planes, and the contact planes are, by definition, the kernel of $$w$$

Now, I have no clue of the other par:, why the transversality of $$R_w$$ means that the flow of $$R_w$$ preserves the form. Any ideas?
Thanks.
 

What are Reeb Fields and Contact Fields?

Reeb Fields and Contact Fields are topological structures used to study the properties of a function or a data set.

What is the difference between Reeb Fields and Contact Fields?

Reeb Fields focus on the level sets of a function, while Contact Fields focus on the gradient of a function. In other words, Reeb Fields analyze the horizontal structure of a function, while Contact Fields analyze the vertical structure.

How are Reeb Fields and Contact Fields useful in scientific research?

Reeb Fields and Contact Fields have various applications in fields such as data analysis, pattern recognition, and computer graphics. They can help identify critical points, understand the topology of a data set, and visualize complex data.

What are some common techniques used to construct Reeb Fields and Contact Fields?

Some common techniques include Morse theory, critical point theory, and gradient descent. These methods involve analyzing the topology of a function or data set to identify important features and construct the corresponding fields.

Are there any limitations to using Reeb Fields and Contact Fields?

Like any scientific tool, Reeb Fields and Contact Fields have their limitations. They may not be applicable to all types of data or functions, and their construction may be computationally expensive. Additionally, their interpretations may be subject to noise and other factors.

Similar threads

  • Differential Geometry
Replies
5
Views
1K
Replies
26
Views
6K
  • Differential Geometry
Replies
9
Views
2K
Replies
1
Views
2K
Replies
5
Views
2K
  • Differential Geometry
Replies
1
Views
2K
  • Differential Geometry
Replies
11
Views
3K
  • Differential Geometry
Replies
2
Views
1K
Replies
3
Views
2K
  • Introductory Physics Homework Help
Replies
11
Views
368
Back
Top