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Orientations of curves and diffeomorphism

  1. Mar 10, 2009 #1

    WWGD

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    Hi:
    I am trying to show that if we have a diffeomorphism f:M-->N and C is

    a positively-oriented Jordan curve in M ( so that., the winding number of C about any

    point in its interior is 1 ) , then f(C) is also positively-oriented in the same sense.


    It seems like something obvious to do is to use the fact that if F : M-->N is a diffeo.

    then F_* T_pM and T_pF(M) is a V.Space isomorphism. I imagine we can consider the

    curve (since it is a Jordan curve, I think reasonably-nice ) as embedded in M , and

    then we can see the tangent space of the curve as a subspace of T_pM , and so we

    have a vector space isomorphism G* T_qM -->T_F(q)M for q in the curve.


    Now, I think we can describe that a curve is positively-oriented by using a V.Field

    (which points towards the interior of C at each point, so that if we are walking along the

    curve, the interior will be to our left) , and the diffeo. inducing a V.Space isomorphism,

    at each point, should preserve this property, but I don't see how to make this more

    precise; I don't even know how to define a V.Field that describes positive orientation.


    Thanks For Any Ideas.
     
  2. jcsd
  3. Mar 11, 2009 #2

    WWGD

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    I realized we cannot talk about Jordan curves without complications (i.e., when

    we have a surface of genus >1 ) unless M,N are surfaces. And even then, not

    every diffeo. is orientation-preserving.
     
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