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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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Shmuel (Seymour J.) Metz

Posts: 3,338
Registered: 12/4/04
Re: How do we Evaluate This Form on S^1?
Posted: Jan 22, 2013 9:07 AM
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In <lbGdnZWOCuemA2DNnZ2dnUVZ5rmdnZ2d@giganews.com>, on 01/21/2013
at 11:59 AM, "W. Dale Hall" <wdhall@alum.mit.edu> said:

>Which statement is the "in fact, false" non-sequitur?

"tangent vectors to different points on S^1 aren't quite identifiable
with one another"

>I maintain that my first statement is perfectly correct.

You can maintain it all you want, but it remains false.

>Otherwise, why would one ever worry about the whole machinery of vector bundles?

Because in the general case you're not dealing with a flat affine
connection.

>A beginner needs to learn that the tangent bundle is something
>other than a vector space;


For which purpose S^2 is mopre suitable than S^1.

>I suspect that the OP knows little to nothing about parallel transport.

It might have been useful to mention it, stressing that what is true
of S^1 is not true in general, even for S^2.

>Again, I regard it as a pedagogical error to use machinery without
>adequate justification


I regard it as a pedagogical error to oversimplify.

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