What Distinguishes a Reeb Vector Field from a General Contact Field?

[WWGD]

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\Phi , and w is the contact form, then:

\Phi^*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 \Phi_* 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 \Psi, then \Psi^*(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 \Psi is a sort of path of contact-form-preserving maps, i.e., for each teach \Psi_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?

 

[Ben Niehoff]

Everything you've written down makes sense. What's your actual question? Do you understand flows and pushforwards?

 

[WWGD]

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.

 

[WWGD]

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

T_pM to T_pM , mapping a deriva

I'm also having trouble understanding or seeing a more geometric interpretation of the result
L_Rw 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(R_w,.)=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_Rω 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.

 

[WWGD]

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.