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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?

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# Reeb Fields and Contact Fields.

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