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Topic: How do we Evaluate This Form on S^1?
Replies: 11   Last Post: Jan 22, 2013 4:17 PM

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Bacle H

Posts: 283
Registered: 4/8/12
Re: How do we Evaluate This Form on S^1?
Posted: Jan 21, 2013 11:05 PM
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On Monday, January 21, 2013 11:59:49 AM UTC-8, W. Dale Hall wrote:
> Shmuel (Seymour J.) Metz wrote:
>

> > In <x5qdnfD8tJNRFmHNnZ2dnUVZ5tidnZ2d@giganews.com>, on 01/20/2013
>
> > at 04:28 PM, "W. Dale Hall" <wdhall@alum.mit.edu> said:
>
> >
>
> >> A tangent vector isn't *really* a real number
>
> >
>
> > That much is true.
>
> >
>
> >> [after all, tangent vectors to different points on S^1 aren't
>
> >> quite identifiable with one another since they belong to different
>
> >> tangent spaces],
>
> >
>
> > That's a non sequitur and, in fact, is false. The standard affine
>
> > connection on S^1 is flat, so parallel transport establishes an
>
> > isomorphism between T_x and T_y, for x and y is S^1.
>
> >
>
>
>
> Which statement is the "in fact, false" non-sequitur? My remark that
>
> tangent vectors to S^1 at different points belong to different tangent
>
> spaces? That one can't "quite" identify tangents at one point to
>
> tangents at another?
>
>
>
> I maintain that my first statement is perfectly correct. Otherwise, why
>
> would one ever worry about the whole machinery of vector bundles? Even
>
> a trivial bundle has distinct fibres over distinct points, and I regard
>
> it as a pedagogical mistake to use the (what used to be called) "abuse
>
> of language" to ignore the distinction between tangent vectors at one
>
> point and tangent vectors at another point. A beginner needs to learn
>
> that the tangent bundle is something other than a vector space; the
>
> OP's question showed at least that level of confusion, and to dispel
>
> it, I felt it necessary to point out where he/she had gone awry.




>
>
>
> As for my second remark, please note my use of the weasel-word "quite".
>
> It was intended to push the OP off the notion that one simply states
>
> (from the original article):
>
>
>
> The tangent space to S^1 is R^1, so that a tangent vector is
>
> of the form c ; any non-zero real number can be a basis.



My mistake, sorry; I meant to say that the tangent space to any point p in S^1
is 1-dimensional, so that this tangent space is ( up to iso.) a copy of the reals. I'm aware of the bundle as the disjoint union of tangent spaces at all points, and of forms as sections of the bundle (1-forms as sections of the bundle; k forms as sections of powers of the bundle) , but I don't know how to use this fact to evaluate d\theta. I imagine that d\theta being a 1-form, there aren't really that many choices for its definition. I would appreciate it if you could explain how to evaluate d\theta from this perspective of sections and trivializations.

You have been very helpful already; I hope I'm not asking too much from you, please let me know if this is so.

Unfortunately, I know some disjointed bits, but I have trouble putting them together. I know the bundle TS^1 is trivial, since, e.g., (-sin(theta),cos(theta)) is a nowhere-zero tangent vector field. I just don't seem to put all this together. I would appreciate your suggestions for help in this respect.
>
>
>
> This statement betrays a fundamental misunderstanding. Note that there
>
> is no mention of the tangent bundle, and without some cognizance of
>
> that object, one can never progress to handle non-trivial problems.
>
>
>
> I sought to correct that misunderstanding. There is NO "tangent space
>
> to S^1" without identification of a point in S^1. As you are no doubt
>
> aware, one correctly identifies the tangent bundle as the union of
>
> tangent spaces AT the various points in S^1, subject to the appropriate
>
> topology. Even in the trivial case of S^1, it is important to pay
>
> attention when using the triviality of the tangent bundle TS^1. For
>
> instance, if the OP had said something like:
>
>
>
> let T : S^1 x R ---> TS^1 be a trivialization of TS^1,
>
>
>
> and proceeded from there to attempt to evaluate the form d(theta) at
>
> an element of TS^1, I'd have argued somewhat differently. I know
>
> nothing of the OP's background, but inferred from his/her level of
>
> discussion that I was addressing a beginner.
>
>
>
> In my opinion, one doesn't simply identify tangents at one point to
>
> tangents at another without some appeal to parallel transport or
>
> (failing the assumption of some connection) triviality of the
>
> restriction of the tangent bundle to the neighborhood of some path
>
> connecting two points. Even then, a carefully-worded argument will
>
> at least employ such an identification via an explicit mapping. I
>
> note that you mentioned parallel transport; I suspect that the OP
>
> knows little to nothing about parallel transport. In a similar (albeit
>
> ridiculous) vein, I could have appealed to a null-homotopy of the
>
> classifying map S^1 ---> BGL(1,R). I don't imagine that would have
>
> been at all useful, either.
>
>
>
> Again, I regard it as a pedagogical error to use machinery without
>
> adequate justification (such as the audience, the context of the
>
> discussion). My take on the OP's article suggested that something
>
> other than "let the real number c be tangent to S^1" needed to be
>
> said.
>
>
>
> I apologize if I misunderstood your remarks.
>
>
>
> Dale





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