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We can define the tangent bundle in two ways, depending on how we define our tangent spaces. If our tangent spaces are derivations on the space of smooth germs at $p \in \mathcal M$ (where $\mathcal M$ is a smooth manifold without boundary), then the tangent bundle is $$T\mathcal M := \bigcup_{p\in \mathcal M} T_p\mathcal M$$ since the $T_p\mathcal M$ are already disjoint under this definition. Alternatively, if we define $T_p\mathcal M$ to be the space of derivations at $p$, then $$T\mathcal M := \coprod _{p \in \mathcal M} T_p\mathcal M =\bigcup_{p\in \mathcal M}\{p\}\times T_p\mathcal M.$$ My questions are:

  1. If we define $T_p\mathcal M$ in the second way, how is it that we can say $T\mathcal M$ is a vector bundle with typical fibre $T_p\mathcal M$?
  2. What are the advantages/disadvantages of each definition?
Daniel Waters
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  • What is the difference between these two definitions, topologically? – Berci Apr 07 '21 at 18:08
  • @Berci, I think they are diffeomorphic. The book I'm reading (Lee's intro to smooth manifolds) uses the second one, but the first one gives a better intuition in terms of $T_p\mathcal M$ being the fibre of $T\mathcal M$. – Daniel Waters Apr 07 '21 at 18:12
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    For the first, fibers are $T_pM$, for the second, fibers are ${p}\times T_pM$, which are basically the same sets. The thing is that the second is the rigorous way to define the disjoint union of sets, while the first is an implicit disjoint union (they are not subsets of the same ambiant space). – Didier Apr 07 '21 at 18:26
  • @Didier What are the advantages of each one? – Daniel Waters Apr 07 '21 at 18:37
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    The second shows explicitely that it is a disjoint union, while the first can be ambiguous in the context of submanifold of $\mathbb{R}^n$. – Didier Apr 07 '21 at 18:39
  • Okay, so when writing it out, emphasizing that the first is a disjoint union would be a good idea? – Daniel Waters Apr 07 '21 at 18:40
  • Yes. But the very definition of a disjoint union is... The second definition you wrote! See this wikipedia article – Didier Apr 07 '21 at 18:41
  • Yes, I know that. How exactly does the germ definition cause problems in the case of submanifolds? – Daniel Waters Apr 07 '21 at 18:44
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    In the setting of submanifolds of $\mathbb{R}^n$, tangent spaces are not usually defined as germs. Of course, you can use some highly-advanced machinery to bypass problems in definitions, but then you loose both the historical idea of the definition and the visual sense. – Didier Apr 08 '21 at 09:54
  • Oh okay. I'll look into that. Thanks for your help! – Daniel Waters Apr 08 '21 at 12:33
  • Derivations at $p$: Applied to germs at $p$ or to smooth maps $\mathcal M \to \mathbb R$? – Paul Frost Apr 09 '21 at 23:38
  • Does this answer your question? Tangent bundle: disjoint union – Paul Frost Apr 09 '21 at 23:41
  • @PaulFrost derivations on the ring (or algebra) of germs at $p$. Your answer doesn't go into $T_p\mathcal M$ as the vector space of derivations on $C_p^{infty}(\mathcal M)$. I have pretty much answered this question myself, I think the first definition is more general (especially on analytic manifolds), and therefore more advantageous. Thanks for your comment! – Daniel Waters Apr 10 '21 at 01:55

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