Orientations of curves and diffeomorphism

In summary, the conversation discusses the relationship between diffeomorphisms and positively-oriented Jordan curves. It is mentioned that a diffeomorphism preserves the positive orientation of a curve, but the exact definition of a vector field that describes positive orientation is unclear. It is also noted that this concept only applies to surfaces and not every diffeomorphism is orientation-preserving.
  • #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.
 
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  • #2
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.
 

1. What is the definition of an orientation of a curve?

An orientation of a curve is a way of assigning a direction to the curve by choosing a starting point and a direction of travel along the curve. It is represented by an arrow pointing in the chosen direction.

2. How is an orientation of a curve related to the concept of a diffeomorphism?

A diffeomorphism is a smooth, bijective function between two smooth manifolds that has a smooth inverse. In the context of curves, a diffeomorphism can be used to transform a curve with a given orientation into a new curve with a different orientation. This transformation preserves the smoothness and bijectivity of the curve.

3. Can a curve have more than one orientation?

No, a curve can only have one orientation at any given point. However, the orientation of a curve can change along its length, depending on the direction of travel chosen at each point.

4. How do orientations of curves relate to the study of differential geometry?

Orientations of curves are an important concept in differential geometry, as they provide a way of studying the behavior of curves and their transformations. They are also used to define concepts such as tangent spaces and tangent vectors, which are crucial in understanding the geometry of curves and surfaces.

5. How are orientations of curves used in practical applications?

Orientations of curves have many practical applications, such as in computer graphics, robotics, and computer-aided design. In these fields, they are used to define the direction of motion of objects and to calculate the orientation of a curve at a given point. They are also used in physics and engineering to study the behavior of particles and systems in motion.

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