Transversality of a Vector Field in terms of Forms (Open Books)

In summary: Let me know if you have any further questions or need additional clarification.In summary, the argument is made using properties of differential forms and open book decompositions of contact 3-manifolds. A contact structure is supported by an open book decomposition if it can be isotoped to have a contact 1-form ω satisfying certain conditions. The equivalence between two conditions is being shown, and the Reeb field R_ω is crucial in this equivalence as it is positively-tangent to the binding B and positively-transverse to the pages of the open book. The notation i_{R_ω}(ω /\ dω) denotes the contraction of the form ω /\ dω by the Reeb field R_ω,
  • #1
WWGD
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Hi, All:
Sorry for the length of the post, but I think it is necessary to set things up so that the post is understandable:

I'm going through an argument in which we intend to show that a given vector field [ itex]R_ω [/ itex]
(actually a Reeb field associated with a contact form ω) is positively-tangent to a
link ( as in a space whose connected components are [itex] S^1 [/itex] -knots. This means the Reeb field
lives in the tangent space to the Link, along the positive direction) , and [itex] R_{\omega}[/itex] is positively-
transverse to a surface S ( so that [itex]R_{\omega}[/itex] intersects S positively at points).
The argument is made using properties of differential forms, in the context of open book decompositions of contact 3-manifolds.
A needed definition is this: A contact structure ζ is _supported by an open book decomposition (B, π )_ if ζ can be isotope thru contact structures so that there is a contact 1-form ω for ζ satisfying:

1)dω is a positive area form on each page [itex]∑_{\theta}[/itex] of the open book, and:

2)ω>0 on the binding B ; both B and the pages are oriented.

End of setup.

------------------------------------------------------------------------------
Actual Question:

To be more specific, I'm trying to understand the following arguments purporting to show
the equivalence between these two conditions:

(1) The contact manifold (M,ζ ) is supported by the open book (B,π)

(3) There is a Reeb field [itex]R_ω[/itex] for a contact structure isotopic to ω , so that [ itex]R_ω[/itex] is positively-tangent to the binding B, and positively-transverse to the pages of the open book.

Proof:
(3)->(1) : Since [itex] R_ω[/itex] is assumed positively -tangent to the binding B , we have ω>0 on oriented tangent vectors to B. Since the Reeb field [itex]R_ω [/itex] is positively-transverse to the pages of the OB (open book) , we have that [itex]dω=i_{R_ω}[/itex](ω /\ dω) >0 on the pages of the OB (where i is --I am? -- the interior product , or contraction of the form ω /\ dω by the vector field
[itex] R_ω[/itex]

Questions:
i)How does [itex]R_{ω}[/itex] being positively-tangent to the ( knots in the ) binding imply ω >0 ?
I know this means the vector field being positively-tangent to the binding means that [itex]R_{ω}[/itex] lies along the tangent space ; a 1-d tangent space, to each of the knots, along the chosen positive direction orientation.

ii)Why is dω equal to the contraction of ω /\ dω ? , and how does the positive transversality imply that dω>0?

(1)->(3): Assume (1), and let ω be the form with the given conditions, and let [itex]R_{ω}[/itex] be the Reeb field associated with ω. Then "It is clear that [itex]R_{ω}[/itex] is positively-transverse to the pages of the OB, since dω is an area form on the pages of the open book"

I have no clue about the connection between the Reeb field being positively-transverse to the pages, and dω being an area form on the pages. I know if [itex]dw[/itex] is a positive area form on the pages, then dω (X,Y)>0 at any pair of positively-oriented tangent vectors. And I know a Reeb field associated to a contact structure is transverse to the planes in the contact structure .But I can't see how this relates to $dω being an area form for the pages of the open book.

Thanks for any suggestions, ideas.
 
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Dear fellow scientist,

Thank you for sharing your question and providing the necessary background information for context. After reviewing the argument and questions, I believe the following explanations may help clarify the connection between the Reeb field and the differential form in question.

i) The statement "R_ω is positively-tangent to the binding B" means that the Reeb field R_ω is tangent to the binding B in a positive direction. This implies that ω>0 on oriented tangent vectors to B because, by definition, the Reeb field is a vector field that is positively-tangent to the binding. In other words, the Reeb field is defined as a vector field that points in the positive direction on the binding, and since ω is a 1-form that assigns a value to each tangent vector, it must also be positive on those vectors.

ii) The notation i_{R_ω}(ω /\ dω) denotes the contraction of the form ω /\ dω by the Reeb field R_ω. This operation essentially takes the value of ω /\ dω at a point and multiplies it by the value of R_ω at that same point. The result is a scalar value, which in this case is dω. The positive transversality of R_ω to the pages of the open book implies that dω is positive on those pages. This is because, as you mentioned, dω is a positive area form on the pages of the open book, and the Reeb field is transverse to those planes. This means that the value of dω at each point is positive, and since the Reeb field is positively-transverse to those points, the contraction of ω /\ dω by R_ω will also be positive.

iii) In the reverse direction, assuming (1) and letting ω be the form with the given conditions, it is clear that R_ω is positively-transverse to the pages of the open book because dω is an area form on those pages. This is because the Reeb field R_ω is transverse to the planes of the contact structure, and since ω is a 1-form that assigns a value to each vector in that plane, the contraction of ω /\ dω by R_ω will be positive. This implies that R_ω is positively-transverse to the pages of the open book.

I hope this helps clarify the connections between the Reeb field and the
 

What is the concept of transversality in vector fields?

Transversality in vector fields refers to the geometric relationship between a vector field and a submanifold. It is a measure of how well the vector field intersects or cuts across the submanifold. A vector field is transverse to a submanifold if it is not tangent to the submanifold at any point.

How is transversality of a vector field related to forms?

In the context of differential geometry, forms are mathematical objects used to study vector fields and their interactions with submanifolds. The concept of transversality is defined in terms of the exterior derivative of a form, which describes the rate of change of the form along the vector field. If the exterior derivative is non-zero, the vector field is transverse to the submanifold.

What are the practical applications of understanding transversality in terms of forms?

Transversality is a fundamental concept in differential topology and geometry, and has many practical applications in physics, engineering, and computer science. It is used to study the behavior of dynamical systems, such as fluid flow, and to analyze the stability and bifurcations of solutions. It is also essential in the study of vector fields on manifolds, which have many real-world applications in fields such as robotics and computer graphics.

How is the concept of transversality related to open books?

An open book is a type of foliation on a manifold, which can be described by a form. The transversality of a vector field with respect to the open book is related to the existence and uniqueness of solutions to a differential equation on the manifold. In particular, when a vector field is transverse to an open book, it guarantees the existence of a unique solution to the differential equation.

What are some techniques used to study transversality of vector fields in terms of forms?

Some commonly used techniques to study transversality include Morse theory, which relates the topology of a manifold to the behavior of a vector field on it, and spectral sequences, which can be used to analyze the algebraic structure of forms and their derivatives. Other techniques include the use of Lie derivatives and the Poincaré lemma to calculate the exterior derivative of forms.

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