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Re: How do we Evaluate This Form on S^1?
Posted:
Jan 21, 2013 2:59 PM
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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.
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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