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Contact Vector Fields. "Flow Preserves Contact Structure?

  1. Sep 17, 2014 #1

    WWGD

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    Hi All, I am going over a definition of a Contact Vector Field defined on a 3-manifold: this is defined as " a vector field v whose flow preserves the contact structure " .
    1) Background (sorry if this is too simple) A contact structure ## \xi ##( let's stick to 3-manifolds for now ) is a nowhere-integrable plane bundle on a 3-manifold M^3, i.e., we have a 2-plane distribution so that there are no submanifolds N < M^3 (i.e., surfaces here) so that TN = ## \xi ## , i.e., there are no submanifolds N of M^3 whose tangent bundle coincides with the contact distribution (this is related to one of Frobenius' theorems and involutivity).

    Now ,does the statement " the flow of the vector field v preserves the contact structure" mean that the tangent space T_C(t) along any flow curve C(t) (local or global) coincides with the contact plane at C(t) ?

    Thanks.
     
  2. jcsd
  3. Sep 19, 2014 #2
    I don't understand the question. The way I would interpret the definition is just that the flow is a local diffeomorphism at each point in time, and, as such, it will map each contact plane to some other plane at the image point. And the definition is requiring contact planes to map to contact planes. At least, that's what it sounds like to me.
     
  4. Sep 21, 2014 #3

    WWGD

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    Well, yes, the pullback of the flow sends contact planes to contact planes, but I wonder if the stronger condition that the tangent planes at/along points in the flow curves are also contact planes. Basically, I am trying to understand the difference between contact vector fields and Reeb fields; contact fields preserve the contact structure, while Reeb fields preserve the contact form (the contact form is a 1-form w whose kernel is the contact distribution); every Reeb field is a contact field, but not necessarily the other way around.

    One difference is given by the Lie derivative of the form w by a Reeb field R , which is 0, while the Lie derivative of w by a contact field V is g.w , where ## g: M^3 \rightarrow \mathbb R ## is a function ( so g==0 gives us a Reeb field ).

    I guess it all comes down to my not having a good way of interpreting the Lie derivative. I do know this is a way of differentiating by approaching a point along the flow of a vector field, but I don't have a clear idea of what having the Lie derivative be 0 or g.w means.
     
  5. Sep 22, 2014 #4
    That's the part I couldn't make sense of. If you just say "tangent plane" with no context, to me that sounds like the whole tangent space of the 3-manifold at those points, which is a 3-dimensional vector space, and the other thing it could be would be the tangent space to the curve, but that's 1-dimensional. I don't see what else is there to be preserved other than the contact planes themselves.
     
  6. Sep 22, 2014 #5

    WWGD

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    Well, looking at the points in the flow curves as points in the 3-manifold, each of these points will be assigned a tangent plane under the contact distribution. These tangent planes will themselves have tangent spaces which are tangent planes , at each point. I don't know if I am too far-off, but can't the tangent map "induced" by the flow (this map is a linear isomorphism) take contact planes to tangent planes to the contact planes?
     
  7. Sep 22, 2014 #6

    WWGD

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    Never mind, you're right, I need to think the question through some more.
     
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