I asked this question yesterday, and edited it today to include clarification.

It is - I think - a fairly honest and straightforward question, and I have provided a detailed explanation of my thinking within the body of the question.

Perhaps I have made mistakes, but as far as I can tell the key points of my explanation are consistent with available literature on differential geometry with the exception of disallowing exceptions to the rule of either distinguishing between or identifying isomorphic structures (the latter of which I believe is more conventional, and so have chosen to employ.) I have highlighted the relevant points:

  • Minimizing complexity

  • The consistent application of the convention of identifying isomorphic structures

  • Why the consistent application of convention matters

  • The chain of implications leading to the independence of the tangent space from smooth structure

  • The distinction between homeomorphism and diffeomorphism

  • The inadmissibility of the tangent spaces distinguished by points while retaining the "intrinsic" nature of the tangent space

  • The possibility of defining the tangent bundle to facilitate the transfer of smooth structure, and the possibility of defining it as a topological invariant

I am quite certain that I am incorrect in my thinking as well, but that is what the question is about, and none of the responses - in comments or the one posted answer - have actually addressed this.

Is there an issue with the content of the question beyond my simply being incorrect about some aspect of differential geometry? Is there an inconsistency in my reasoning? Is it my tone? How can I improve the question?

  • $\begingroup$ As far as I can tell, you are defining an object which is not the usual definition of tangent plane/tangent bundle (in the sense that when $M$ is a smooth manifold, your definition is not the one defined in any standard text). Since there are clear geometrically meaning and wide applications with the usual definition of tangent bundle, I think the burden is on you to convince us why your definition is better, not the other way round. $\endgroup$ Dec 2, 2023 at 2:48
  • $\begingroup$ @ It isn't "better," and it isn't supposed to be. It's meant to be equivalent but simpler. I derived the definition by sitting down with a list of agreed upon properties of tangent space and tangent bundles and and saying "how can I make all of these true in the easiest way possible." It was my intention that my definition agree with the specific constructions of the tangent space in terms of equivalence classes of curves and derivations wherever we these are defined, but I wasn't sure, so I asked to see if I was missing something. $\endgroup$
    – R. Burton
    Dec 2, 2023 at 18:18
  • $\begingroup$ @ArcticChar In the body of the question, I included a means of ensuring that the tangent space exist only for smooth manifolds. I found this construction to be pointless, since it adds complexity without contributing anything but the nominal "and also this happens to be a smooth manifold." I expected that if there is something more than that, someone would explain what that something is (up to isomorphism). $\endgroup$
    – R. Burton
    Dec 2, 2023 at 18:20
  • $\begingroup$ @ArcticChar "What is wrong with my question ..." : my (subjective) view is that the quality of the MathSE question is not to be evaluated by the accuracy/appropriateness of any assertions made in the posted question. That would be equivalent to down-voting a well written Math problem-question that shows much work, simply because the OP made analytical errors. To me, the issue is: did the presented question make a sincere attempt to provide context, show work, and present ideas in a reasonably composed manner. ...see next comment $\endgroup$ Dec 4, 2023 at 19:25
  • $\begingroup$ @ArcticChar Personally, I regard the whole situation as yet another example that there is no quality control on down-voting. For what it's worth, I have noticed a pattern to such down-voting. It seems to be more frequent when the posted question is inconvenient/difficult to confront. MathSE reviewers aren't androids, and they are inundated with (for example) low quality PSQ's. So, it is perhaps not surprising when the reviewer blurs the distinction between one type of negative emotional reaction and another type of negative emotional reaction. $\endgroup$ Dec 4, 2023 at 19:31


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