Bacle H
Posts:
188
Registered:
4/8/12
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Re: How do we Evaluate This Form on S^1?
Posted:
Jan 21, 2013 11:18 PM
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On Monday, January 21, 2013 8:05:53 PM UTC-8, bacle...@gmail.com wrote:
> On Monday, January 21, 2013 11:59:49 AM UTC-8, W. Dale Hall wrote:
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> > Shmuel (Seymour J.) Metz wrote:
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> > > In <x5qdnfD8tJNRFmHNnZ2dnUVZ5tidnZ2d@giganews.com>, on 01/20/2013
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> > > at 04:28 PM, "W. Dale Hall" <wdhall@alum.mit.edu> said:
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> > >> A tangent vector isn't *really* a real number
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> > > That much is true.
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> > >> [after all, tangent vectors to different points on S^1 aren't
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> > >> quite identifiable with one another since they belong to different
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> > >> tangent spaces],
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> > > That's a non sequitur and, in fact, is false. The standard affine
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> > > connection on S^1 is flat, so parallel transport establishes an
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> > > isomorphism between T_x and T_y, for x and y is S^1.
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> > Which statement is the "in fact, false" non-sequitur? My remark that
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> > tangent vectors to S^1 at different points belong to different tangent
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> > spaces? That one can't "quite" identify tangents at one point to
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> > tangents at another?
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> > I maintain that my first statement is perfectly correct. Otherwise, why
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> > would one ever worry about the whole machinery of vector bundles? Even
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> > a trivial bundle has distinct fibres over distinct points, and I regard
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> > it as a pedagogical mistake to use the (what used to be called) "abuse
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> > of language" to ignore the distinction between tangent vectors at one
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> > point and tangent vectors at another point. A beginner needs to learn
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> > that the tangent bundle is something other than a vector space; the
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> > OP's question showed at least that level of confusion, and to dispel
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> > it, I felt it necessary to point out where he/she had gone awry.
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> > As for my second remark, please note my use of the weasel-word "quite".
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> > It was intended to push the OP off the notion that one simply states
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> > (from the original article):
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> > The tangent space to S^1 is R^1, so that a tangent vector is
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> > of the form c ; any non-zero real number can be a basis.
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> My mistake, sorry; I meant to say that the tangent space to any point p in S^1
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> 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.
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> You have been very helpful already; I hope I'm not asking too much from you, please let me know if this is so.
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> 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.
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> > This statement betrays a fundamental misunderstanding. Note that there
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> > is no mention of the tangent bundle, and without some cognizance of
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> > that object, one can never progress to handle non-trivial problems.
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> > I sought to correct that misunderstanding. There is NO "tangent space
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> > to S^1" without identification of a point in S^1. As you are no doubt
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> > aware, one correctly identifies the tangent bundle as the union of
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> > tangent spaces AT the various points in S^1, subject to the appropriate
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> > topology. Even in the trivial case of S^1, it is important to pay
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> > attention when using the triviality of the tangent bundle TS^1. For
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> > instance, if the OP had said something like:
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> > let T : S^1 x R ---> TS^1 be a trivialization of TS^1,
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> > and proceeded from there to attempt to evaluate the form d(theta) at
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> > an element of TS^1, I'd have argued somewhat differently. I know
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> > nothing of the OP's background, but inferred from his/her level of
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> > discussion that I was addressing a beginner.
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> > In my opinion, one doesn't simply identify tangents at one point to
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> > tangents at another without some appeal to parallel transport or
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> > (failing the assumption of some connection) triviality of the
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> > restriction of the tangent bundle to the neighborhood of some path
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> > connecting two points. Even then, a carefully-worded argument will
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> > at least employ such an identification via an explicit mapping. I
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> > note that you mentioned parallel transport; I suspect that the OP
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> > knows little to nothing about parallel transport. In a similar (albeit
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> > ridiculous) vein, I could have appealed to a null-homotopy of the
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> > classifying map S^1 ---> BGL(1,R). I don't imagine that would have
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> > been at all useful, either.
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> > Again, I regard it as a pedagogical error to use machinery without
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> > adequate justification (such as the audience, the context of the
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> > discussion). My take on the OP's article suggested that something
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> > other than "let the real number c be tangent to S^1" needed to be
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> > said.
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> > I apologize if I misunderstood your remarks.
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> > Dale
Sorry, hbert, and all, I confused this with one of my posts.
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