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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 coordinate-independent 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

of which is γ(t)=(cost,sint), and the other is the polar-coordinate expression:

γ

Now, in the first parametrization, the velocity is given by:

γ'(t)=(-sint, cost) , while

in the second one, we get:

γ'

So, in what sense is the velocity coordinate-independent 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

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 naturally-isomorphic to each other, but I do not see anywhere

in this chapter where/how those isomorphisms are defined. Any Ideas/Suggestions?

Thanks.

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 coordinate-independent 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}, oneof which is γ(t)=(cost,sint), and the other is the polar-coordinate 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 coordinate-independent 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)]/Δtthe 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 naturally-isomorphic to each other, but I do not see anywhere

in this chapter where/how those isomorphisms are defined. Any Ideas/Suggestions?

Thanks.

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