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" <email@example.com> 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