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Velocity Vector has CoordinateIndependent Meaning 
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#1
Mar2912, 07:56 PM

P: 513

Hi, All:
It's been a long time since I did this, and I have some basic doubts; please bear with me: In Lee's Riemannian Mflds, p.48, he states that , given a parametrization : γ:(a,b)→M, of a curve , "the velocity vector γ'(t) has a coordinateindependent meaning for each t in M" (this should be for each t in (a,b). Now, Lee goes on to give an example of two parametrizations of S^{1} , one of which is γ(t)=(cost,sint), and the other is the polarcoordinate expression: γ _{2}(t)=(r(t),θ(t))=(1,t) . Now, in the first parametrization, the velocity is given by: γ'(t)=(sint, cost) , while in the second one, we get: γ'_{2}(t)=(0,1) So, in what sense is the velocity coordinateindependent then? Thanks. EDIT: Moreover, the reason given for the "ambiguity" in defining acceleration seems to apply to the definition of velocity too: In the difference quotient Lim_{Δt→0} [f(x+Δt)f(x)]/Δt the vectors x and x+Δt live in tangent spaces that are not naturally isomorphic to each other, right? Lastly hope this is not too long of a question I understand at an informal level that a connection is a device used to define/select a choice of isomorphism between vector spaces that are not naturallyisomorphic to each other, but I do not see anywhere in this chapter where/how those isomorphisms are defined. Any Ideas/Suggestions? Thanks. 


#2
Mar2912, 08:10 PM

P: 513

Moreover, it would be great if someone could guide me thru the case of the circle itself,
on how to define a connection on it. I usually learn more from working with coordinates and then doing the abstracting myself, than from seeing the material abstracted by someone else without my knowing what is being abstracted. Anyway, Thanks. Just to note that I read PRof. Quasar and Others' recent post on Connections, but it was stated there that the post applied to principal bundles, and not to vector bundles. 


#3
Apr112, 01:18 PM

P: 49

I'm just learning this stuff myself, so please correct me if I'm wrong.
The velocity vector is a coordinateindependent object that is an element of the tangent space of the point you're considering. But like all vector spaces, there is not a unique choice of basisone usually picks the basis induced by the coordinate lines, ie the partial derivative operators with respect to those coordinates, although you can modify them however you want. So although you derived two different sets of components for the velocity vector, you have (probablyI didn't check your math) derived the same velocity vector. Components specify the weights put on each basis vector so the same vector will generally have different components in a different basis. What always confused me is when people used "change of coordinates" and "change of basis" interchangeably. Really it's the change of basis that matters, and the basis we're talking about is the basis of the tangent space. A change of coordinates will induce a change of basis (if we want it to), but we can consider the basis induced by a particular coordinate system while using a different coordinate system to label points. 


#4
Apr212, 05:44 PM

P: 513

Velocity Vector has CoordinateIndependent Meaning
Ah, good point. I'll try as an exercise to doublecheck to see if there is an actual
(covariant) changeofbasis taking us from (sint,cost), to (0,1). Still, I have not figured out the second issue: how is it that these connections allow us to overcome the issue of vector spaces at different points (say, when we have a vector field along a curve) where the tangent spaces are not naturally isomorphic at each other. I know in R^n ( by def. I think) vector fields in different tangent spaces are parallel iff they have the same component. Once we start with moving frames , e.g., Frenet Serret frames, then the basis frames are no longer parallel , then their displacement has to be taken into account. I'll repost when I figure it out. 


#5
Apr212, 06:04 PM

Sci Advisor
P: 2,652

A connection is the same as specifying a rule for parallel transport. You can specify that rule in some slightly arbitrary way on your manifold (the arbitrariness is limited by a few properties that a connection must have).
You could, for example, choose to embed your manifold in a higher dimensional flat manifold (e.g. 2 sphere in R^3) and parallel transport by first parallel transporting in the flat manifold as usual and then projecting the resulting vector (assuming you have a metric) onto your embedded submanifold. The result of this construction is the socalled LeviCivita connection (one can show that this definition is independent of choice of embedding). The difference between the flatmanifold case is that on a curved manifold, whether 2 vectors are parallel or not depends on the path through which you connect them. 


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