Mathematics Asked by Tug Witt on December 12, 2020

From my understanding of (smooth) manifolds, all you need is an atlas to describe a manifold. However, if you have some atlas ?={($U_n$,$phi_n$)} with $n$ charts, we still haven’t defined our transition maps. My questions are:

- Are the transition maps implied within the atlas (i.e. you can derive all transition maps from a given atlas) or do we have to store our transition maps along with our atlas to prove we have a smooth atlas?
- If you have $n$ charts within an atlas, does that mean you are going to have something like $n!$ (maybe its a bit more complex than
*that*) transition maps? For example, if $n=3$ and a chart $cin A$, wouldn’t you need a transition map from $c_1 -> c_2$, $c_1 -> c_3$, $c_2 -> c_3$ plus all of the inverses (that are implied)? When**don’t**you need a transition map between two charts in the same atlas?

You can simply *define* the transition maps, once the atlas is given.

There is a transition map which I shall denote $psi_{m,n}$ for every pair of indices $m,n$ having the property that $U_m cap U_n ne emptyset$.

The domain of $psi_{m,n}$ is the set $phi_m(U_m cap U_n) subset mathbb R^k$ (I'm assuming implicitly that $k$ is the dimension of the manifold).

The range (or codomain) of $psi_{m,n}$ is the set $phi_n(U_m cap U_n) subset mathbb R^k$.

And the formula for $psi_{m,n} : phi_m(U_m cap U_n) to phi_n(U_m cap U_n)$ is $$psi_{m,n}(p) = phi_n(phi^{-1}_m(p)), quad p in phi_m(U_m cap U_n) $$

Also, once all of this is written down, one can use the definition of a manifold together with the Invariance of Domain Theorem to prove that the domain and range of $phi_{m,n}$ are both open subsets of $mathbb R^k$, and one can show that $psi_{n,m}$ is an inverse map of $psi_{m,n}$, hence each transition map is a homeomorphism from its domain to its range.

And once *that* is done, you can now ask yourself questions that are aimed at determining whether your manifold is a $C^infty$ manifold, or a $C^2$ manifold, or a $C^1$ manifold or whatever smoothness property you want. Namely: Are the functions ${psi_{m,n}}$ all $C^infty$? or are they all $C^2$? or $C^1$?

Correct answer by Lee Mosher on December 12, 2020

Once you have the charts $phi_n$, the transition maps are determined, as $phi_mcircphi_n^{-1}$. (That uses my favorite convention for the direction of these maps; you might need to move the "inverse" if your convention is different.)

Answered by Andreas Blass on December 12, 2020

1 Asked on December 5, 2021 by mr-n

hypergeometric function oeis power series reference request sequences and series

3 Asked on December 5, 2021

1 Asked on December 5, 2021 by wei-xia

2 Asked on December 5, 2021 by yunfei

1 Asked on December 5, 2021 by nothingone

1 Asked on December 5, 2021

1 Asked on December 5, 2021 by kevin-lu

1 Asked on December 5, 2021

2 Asked on December 5, 2021 by user117375

1 Asked on December 5, 2021

0 Asked on December 5, 2021

1 Asked on December 5, 2021 by bigbear

1 Asked on December 3, 2021 by mathguy1345

2 Asked on December 3, 2021 by shad0w7

2 Asked on December 3, 2021 by naren-manoj

eigenvalues eigenvectors linear algebra matrices perturbation theory symmetric matrices

1 Asked on December 3, 2021 by kevinkayaks

0 Asked on December 3, 2021 by kyary

Get help from others!

Recent Answers

- Jon Church on Why fry rice before boiling?
- haakon.io on Why fry rice before boiling?
- Joshua Engel on Why fry rice before boiling?
- Peter Machado on Why fry rice before boiling?
- Lex on Does Google Analytics track 404 page responses as valid page views?

© 2022 AnswerBun.com. All rights reserved. Sites we Love: PCI Database, MenuIva, UKBizDB, Menu Kuliner, Sharing RPP