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Added to tangent bundle the discussion in the context of synthetic differential geometry.
In that context I also restructured a bit and expanded the introduction slightly.
(…almost 8 years later…)
I have started to fill in at Definitions in ordinary differential geometry – Geometric definition details of the classical construction.
Continued to spell out traditional elementary detail at Geometric definition. In particular more of a proof now that the tangent bundle of a differentiable manifold is itself a manifold.
I have re-arranged the sections at tangent bundle:
made all the various sections that existed subsections of the “Definition”-section (because all discuss alternative definitions)
merged what used to be three sections for “algebraic”, “geometric” and “physics” definition (this was not my idea) into a single section “Traditional definition”
(the “algebraic definition” via derivations is one of vector fields, not of the tangent bundle itself, hence hardly an alternative definition; and the “physics” definition via gluing is really the only definition there is: even if one describes the topology on $T X$ in a way that it does not explicitly mention the gluing construction, it is the corresponding quotient topology and the gluing construction is arguably the most transparent way to understand that topology )
Added:
For manifolds of the class $C^k$, $0\lt k\lt \infty$, the definition of a tangent vector as a derivation of the algebra of functions remains valied if one strengthens the definition of a derivation: we must now require
$D(f(g_1,\ldots,g_m))=\sum_i (\partial_i f)(g_1,\ldots,g_m)D(g_i).$See Newns and Walker \cite{NewnsWalker}.
An early account of tangent vectors as derivations, including the $C^k$-case for $0\lt k\lt \infty$ is in
I have added pointer from your remark back to derivations of smooth functions are vector fields. For usability, your remark should probably go to that entry or at least be kept together with the corresponding section here in this entry?
Maybe we could clarify in words what the strengthening is: I guess the point is that instead of demanding just the product rule on a derivation, we also demand the chain rule. If that’s the case, I suggest we state it that way, because otherwise the reader (like me in this case) might be staring at that formula wondering if there is some fine print they are meant to pick up.
Are you planning to create entries for Newns and Walker?
I guess the point is that instead of demanding just the product rule on a derivation, we also demand the chain rule.
Well, yes, but this is the algebraic analogue of the chain rule. In the ordinary chain rule you have derivatives on the right side, here you have an abstract derivation instead.
I created entries for Newns and Walker (the later one clearly deserved an entry).
What is the reason the software accidentally creates empty pages for missing entries? Presumable, there is a rogue hyperlink somewhere? Can we just remove it from the HTML code and forget about this problem?
Okay, I have given the entry a new subsection “Via derivations of smooth functions” (here) and moved into that both the remark to this effect that used to be in the entry, as well as your addendum.
Also, I highlighted that the condition added is the algebraic analogue of the chain rule.
Also, I changed yout
\cite{AuthorNames}
to
[AuthorNames](#AuthorNames)
because the link didn’t work.
What is the reason the software accidentally creates empty pages for missing entries?
I don’t know what this is referring to. Looking at the list of latest revisions doesn’t seem to show anything strange?
I don’t know what this is referring to. Looking at the list of latest revisions doesn’t seem to show anything strange?
This was in reference to your query about Newns and Walker, which I now created. I recall that you once mentioned that it is bad to leave missing entries, since the software tends to accidentally create blank pages for such links.
Hm, I would say: on the contrary, I find it bad to leave broken links because they look bad in the page. They essentially look like an error message, and I find our readers should not be bothered by this.
In fact, I would be happy (happier, at least :-) if the software did in fact create an empty page instead of showing broken links.
[ never mind ]
There may be a typo in Remark 2.14: In the third equation in this section it says $Def(C^\infty(X))$ but it should probably be $Der(C^\infty(X))$ which is the notation used for the derivations of $C^\infty(X)$ at derivations of smooth functions are vector fields. (Martin Biehl)
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