A few questions on manifolds.

  • #1
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In the attached pdf file i have a few questions on manifolds, I hope you can be of aid.
I need help on question 1,2,6,7.
here's what I think of them:
1.
a) the definition of a smooth manifold is that for every point in M we may find a neighbourhood W in R^k of it which the intersection W with M is diffeomorphic to an open neighbourhood U in R^m.
Now if we take a connected component of M, say V, if it intersects W then the restricition of the above diffeomorphism will do.
what do you think of this?
b) I don't think it's irrelevant that M is even a manifold, cause if M is compact and its connected components are disjoint subsets which are connected which their union is M, then obviously if we take some open covering of M then by compactness there's a finite covering of M, this also covers each of its componets, or we may assume that each component has some covering and unite them, it's obviously a covering of M and thus by compactness have a finite covering.

2.I think we need to show this inductively, or better way, if we define
M=U(MnW_i) where the union runs through i, MnW_j is the intersection of M with W_j, such MnW_j is diffeomorphic to some open neighbourhood in R^m.
now we may increase the W_j as big as we please, and thus get increasing sets and if we take the closures of W_j's then those still cover M and are compact in M.
not sure if that will work though.

on questions 6 and 7 I'll ask later perhaps tommorrow or the day after that.
 

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  • #2
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1 a) Your answer sounds reasonable, but you haven't justified WHY when you restrict the diffeomorphism on the entire manifold to a connected component it forms a diffeomorphism on the connected component.
1 b) Sounds good.

2. I'm pretty sure you can't use induction here (prove me wrong if you can). Why can you increase W_j as big as you please? If this were the case you could cover any compact manifold with a single chart. Is this true?
(Hint: Think about circles)
 

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