- #1
- 7,314
- 11,126
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.
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.