0
$\begingroup$

Let $C$ be a $1$-dimensional compact differentiable manifold with boundary in $\mathbb{R}^{3}$.

In easy examples, it looks like we can always parametrize such a manifold with only two charts: usually one that covers almost everything and a last one to cover a missing point.

Is this always true? Is there any theorem that bounds the quantity of minimum charts for a manifold?

I ask this because I am doing a homework about work (physics) and I'm not sure how I would integrate over a manifold that has more than one chart. (I think that when there are two as above, we can just integrate using one chart because the other one is only a point so has measure zero).

Thanks.

$\endgroup$
  • 1
    $\begingroup$ The normal way you integrate over manifolds is with an open cover and a partition of unity. But for 1-manifolds, note that every 1-manifold is diffeomorphic to the circle or $[0,1]$. $\endgroup$ – user98602 Nov 8 '15 at 16:16
  • $\begingroup$ I didn't really study integration over manifolds, only the line integral case, where they use only one chart :(. Cool! Is there a name for such result? $\endgroup$ – Shoutre Nov 8 '15 at 16:18
  • 1
    $\begingroup$ Sure, I was letting you know. The above result means (by the logic in your question) that you do only need one chart. It's the classification of 1-manifolds. $\endgroup$ – user98602 Nov 8 '15 at 16:19
  • $\begingroup$ One chart to integrate, correct? For example, for the line segment joining $(1,2,3)$ to $(4,5,6)$ (containing these points) I would need two charts to parametrize, right? $\endgroup$ – Shoutre Nov 8 '15 at 16:21

Your Answer

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

Browse other questions tagged or ask your own question.