Orientations of curves and diffeomorphism

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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.
 

Answers and Replies

  • #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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