Coordinate systems vs. Euclidean space

[JTC]

Good Morning

I am having some trouble categorizing a few concepts (I made the one that is critical to this post to be BOLD)

1. Remote parallelism: the ability to move coordinate systems and frames around in space.
   
2. Euclidean Space
3. Coordinate systems: Cartesian vs. cylindrical

I am aware that if space is Euclidean, then certain geometric rules hold...

For example, we convert the dot product (from its definition of the product of the norms of two vectors times the cosine of the angle between them) to an algebraic one (where we sum the products of the associated vector components.); and other things about angles to 180, parallel lines, etc.

Distinct from this, is the fact that one can move coordinate systems around. And I have understood this process to be "remote parallelism." And we can do it when the coordinate system is Cartesian. We cannot do it when the system is, say, polar or cylindrical.

So, this is where i need help. I cannot see how Euclidean space is a requirement for remote parallelism. I can see that we need Cartesian coordinates to move frames around. But I cannot see why a Euclidean assumption is required.

 

[Orodruin]

Unfortunately, you have gotten the concepts backwards. As long as the space is Euclidean there is a prescription to move vectors around and a unique definition of parallelism (it is just that it is not just using the same components in coordinates in the Cartesian setting - the components would change). If the space is not flat, then parallelism is not well defined and the parallel transport of a vector from one point to the other will depend on the path. I am sure you have seen the example of the parallel transport of a vector along the sphere.

It should also be noted that it is not necessary for a space to be Euclidean for there to exist a global notion of parallelism. However, the space does need to be flat. A non-trivial example would be the flat torus.

 

[JTC]

OK, I see your point.

However, my understanding of remote parallelism (and I admit I could have this wrong, too), is that if I have a polar coordinate system and move the vector to a new point, then the components of the vector will change (because the radial and theta directions are different at different points). But if it is Cartesian, the components will not change.

Could you please extend your post to this case?

 

[JTC]

Unfortunately, you have gotten the concepts backwards. As long as the space is Euclidean there is a prescription to move vectors around and a unique definition of parallelism (it is just that it is not just using the same components in coordinates in the Cartesian setting - the components would change). If the space is not flat, then parallelism is not well defined and the parallel transport of a vector from one point to the other will depend on the path. I am sure you have seen the example of the parallel transport of a vector along the sphere.
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It should also be noted that it is not necessary for a space to be Euclidean for there to exist a global notion of parallelism. However, the space does need to be flat. A non-trivial example would be the flat torus.

I guess it depends on EXACTLY what "remote parallelism" means (for I have only found conflated definitions).

Does it mean that you can move vectors around (in that case yes, I had it backwards)
Or
Does it mean that if you move the frame around, then the vector's components do not change.

 

[Orodruin]

However, my understanding of remote parallelism (and I admit I could have this wrong, too), is that if I have a polar coordinate system and move the vector to a new point, then the components of the vector will change (because the radial and theta directions are different at different points). But if it is Cartesian, the components will not change

Yes, the components of the vector will generally change. This should not surprise you as the vectors belong to different tangent spaces. The point is that you can do the identification at all (which is a highly non-trivial statement as the vectors belong to different tangent spaces). On a curved manifold, there is no unique way to relate vectors in different tangent spaces.

For a manifold with a given affine connection $\nabla$, a vector field $X$ is parallel if $\nabla_Y X = 0$ everywhere for all $Y$. If there exists a set of parallel vector fields $X_i$ that everywhere form a basis for the tangent space, then you can define a global notion of parallelism in relation to this basis. This is the case for a Euclidean space as the basis vectors in a Cartesian coordinate system forms such a set. However, this does not mean that the notion of parallelism is any different in a different set of coordinates (it is about the vector fields themselves, not about their components in some arbitrary basis).

 

[JTC]

Yes, the components of the vector will generally change. This should not surprise you as the vectors belong to different tangent spaces. The point is that you can do the identification at all (which is a highly non-trivial statement as the vectors belong to different tangent spaces). On a curved manifold, there is no unique way to relate vectors in different tangent spaces.

For a manifold with a given affine connection $\nabla$, a vector field $X$ is parallel if $\nabla_Y X = 0$ everywhere for all $Y$. If there exists a set of parallel vector fields $X_i$ that everywhere form a basis for the tangent space, then you can define a global notion of parallelism in relation to this basis. This is the case for a Euclidean space as the basis vectors in a Cartesian coordinate system forms such a set. However, this does not mean that the notion of parallelism is any different in a different set of coordinates (it is about the vector fields themselves, not about their components in some arbitrary basis).

I still follow your words here, too.

But I am still flummoxed by this definition of "remote parallelism." It now seems to me that it is inextricably tide to the definition of Euclidean space -- so much that they are practically different names for the same issue.

Can you define both of these terms (Euclidean Space vs. Remote Parallelism) so that they are distinct?

Or is one a physical phenomena (remote parallelism) that results from a mathematical/geometric assumption (Euclidean Space)?

 

[Orodruin]

Can you provide references for where you have read about parallelism?

It now seems to me that it is inextricably tide to the definition of Euclidean space -- so much that they are practically different names for the same issue

No, they are definitely not the same. As I said in the previous post, there are many examples of flat manifolds that are not Euclidean spaces, such as the flat torus. However, locally they will look like a Euclidean space. You can also have manifolds that are flat, but neither Euclidean nor have a global notion of parallelism, such as a cone with the apex removed.

Or is one a physical phenomena (remote parallelism) that results from a mathematical/geometric assumption (Euclidean Space)?

Both concepts are mathematical.

 

[JTC]

Can you provide references for where you have read about parallelism?
Both concepts are mathematical.

It is again a case where I read it awhile ago, and no longer remember where. And this compels me to wonder about it now. (I have a vague feeling it was in O'Neil's book)

So, yes, I get that a sphere is not Euclidean. And once identifying the poles, we can say that it is LOCALLY Euclidean. Sure, I get that.

So now we have a patch on the sphere where, assuming locally Euclidean, I get all these ideas about angles in a triangle summing to 180, etc.

But then I apply a coordinate system. If the coordinate system is polar, then when I move a vector around, the components change.
If it is Cartesian, they don't. I thought THAT was the definition of "remote parallelism."

 

[Orodruin]

So, yes, I get that a sphere is not Euclidean. And once identifying the poles, we can say that it is LOCALLY Euclidean. Sure, I get that.

A sphere is not flat. There is no way to uniquely identify the poles. The sphere looks the same everywhere.

But then I apply a coordinate system. If the coordinate system is polar, then when I move a vector around, the components change.
If it is Cartesian, they don't. I thought THAT was the definition of "remote parallelism."

No, definitely not. Anything that depends on your coordinates is not an intrinsic property of the manifold itself. This includes vector and tensor components. They are not a property only of the vector/tensor itself, but also depend on the coordinates that you happen to use. On the contrary, a vector field being parallel has nothing to do with any coordinate system. It only depends on the vector field itself and the affine connection.

 

[JTC]

On the contrary, a vector field being parallel has nothing to do with any coordinate system. It only depends on the vector field itself and the affine connection.

OK; so then this is it and final.

Remote parallelism concerns the ability to move the vector around (and then I can follow your statements)
Remove parallelism is NOT related to moving the frame around and getting the same components.