I have a minor problem on understanding the definition. Say, in the book An Introduction of Manifolds by Loring Tu, he gave a definition as below. My question is: are the chart $(U,\phi)$ and $(V,\psi)$ mentioned in the definition belongs to the atlas of the given manifold $N$ and $M$? To be clear, we know that when we say "manifold $N$", we mean a underlying set together with a "maximal atlas $\Phi_N$," so is $\phi\in\Phi_N$ here? It's hard to guess from the context. enter image description here

Also, Prop 6.8(ii)(iii) is weird to me:

enter image description here


1 Answer 1


Yes: if $M$ is a smooth manifold, then whenever you talk about a chart in $M$ (or on $M$, or of $M$, etc.) that always refers to a chart in the atlas of $M$ unless specified otherwise.

  • $\begingroup$ but then the equivalence of ii and iii in the prop seems too trivial, and it is strange to state as two different sentences $\endgroup$
    – Eric
    Jul 13, 2019 at 17:45
  • 2
    $\begingroup$ The equivalence of (ii) and (iii) is not trivial. (iii) refers to every chart in the atlas of definition of $M$ (i.e., the maximal atlas) while (ii) refers to only charts in some atlas (which may be a proper subset of the maximal atlas). $\endgroup$ Jul 13, 2019 at 17:50

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.