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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 S^1 -knots. This means the Reeb field
lives in the tangent space to the Link, along the positive direction) , and R_{\omega} is positively-
transverse to a surface S ( so that R_{\omega} 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 ∑_{\theta} 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 R_ω 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 R_ω is assumed positively -tangent to the binding B , we have ω>0 on oriented tangent vectors to B. Since the Reeb field R_ω is positively-transverse to the pages of the OB (open book) , we have that dω=i_{R_ω}(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
R_ω
Questions:
i)How does R_{ω} 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 R_{ω} 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 R_{ω} be the Reeb field associated with ω. Then "It is clear that R_{ω} 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 dw 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.
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 S^1 -knots. This means the Reeb field
lives in the tangent space to the Link, along the positive direction) , and R_{\omega} is positively-
transverse to a surface S ( so that R_{\omega} 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 ∑_{\theta} 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 R_ω 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 R_ω is assumed positively -tangent to the binding B , we have ω>0 on oriented tangent vectors to B. Since the Reeb field R_ω is positively-transverse to the pages of the OB (open book) , we have that dω=i_{R_ω}(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
R_ω
Questions:
i)How does R_{ω} 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 R_{ω} 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 R_{ω} be the Reeb field associated with ω. Then "It is clear that R_{ω} 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 dw 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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