Thanks for the detailed explanations of the important topic leading to intrinsic metric on Manifolds. Now we understand that there is a homeomorphism between the (abstract) tangent space at any element of the manifold and R^n. Moreover a Manifold is defined to feature a homeomorphism between any open set and an open set in R^n. Is it now obvious (or can be proven) that a combination of exactly the multiple homeomorphisms you mention related to an element of the Manifold and it's tangent space and then R^n is a homeomorphism between a (or any ? ) neighborhood of that element and an appropriate neighborhood in R^n, and hence can be used in the original definition of the Manifold ?
Maybe it makes sense rephrasing the question with (hopefully) more correct terminology: Can be proven that any map in the atlas of a manifold (the map being associated with a specific point of the manifold) is a combinations of the homeomorphisms (shown in the video) that map the tangent space (for the same point) to R^n ? Afterthought: the map is a *local* homeomorphism, while the Tangent space does not (obviously) even have a topology. Goal: getting rid of one of the multiple homormorphisms we need to deal with for any point.
Tbh i had to rewatch the video multiple times, but slowly it all starts to make sense :D You have any good literature you recommend rereading about these topics? Wikipedia seems sometimes a bit "too much".
Just a naive question as I'm trying to get a better understanding of this material : since the tangent space is a vector space, it is globally Euclidean (isn't it ?), so I don't see why we need to consider a specific chart (U,h) in M around every point p. Shouldn't there be a (non-unique) global chart which maps every tangent space to R^n ? And then we can define the coordinate basis in the same way with respect to that one chart. Why is that not valid ?
@@brightsideofmaths thanks for the reply ! no what I meant is could we "forget" the manifold M and just consider the tangent space TpM, which is already a vector space, to define a basis without using charts from the atlas of M. But I realized that we need this construction of the coordinate basis if we want to do meaningful calculations on the manifold anyway.
@9:47 I’m pretty sure that TpN should really be T_f(p)N or TqN where q=f(p).
Totally right! Sorry for the confusion!
Thank you so much, best mathematics UA-cam channel for mathematics students!!!
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Thanks for the detailed explanations of the important topic leading to intrinsic metric on Manifolds.
Now we understand that there is a homeomorphism between the (abstract) tangent space at any element of the manifold and R^n.
Moreover a Manifold is defined to feature a homeomorphism between any open set and an open set in R^n.
Is it now obvious (or can be proven) that a combination of exactly the multiple homeomorphisms you mention related to an element of the Manifold and it's tangent space and then R^n is a homeomorphism between a (or any ? ) neighborhood of that element and an appropriate neighborhood in R^n, and hence can be used in the original definition of the Manifold ?
Maybe it makes sense rephrasing the question with (hopefully) more correct terminology:
Can be proven that any map in the atlas of a manifold (the map being associated with a specific point of the manifold) is a combinations of the homeomorphisms (shown in the video) that map the tangent space (for the same point) to R^n ?
Afterthought: the map is a *local* homeomorphism, while the Tangent space does not (obviously) even have a topology.
Goal: getting rid of one of the multiple homormorphisms we need to deal with for any point.
Tbh i had to rewatch the video multiple times, but slowly it all starts to make sense :D You have any good literature you recommend rereading about these topics? Wikipedia seems sometimes a bit "too much".
Thanks! We always discuss suitable literature in the community forum. Maybe you can take a look there.
Just a naive question as I'm trying to get a better understanding of this material : since the tangent space is a vector space, it is globally Euclidean (isn't it ?), so I don't see why we need to consider a specific chart (U,h) in M around every point p. Shouldn't there be a (non-unique) global chart which maps every tangent space to R^n ? And then we can define the coordinate basis in the same way with respect to that one chart. Why is that not valid ?
Thank you for the question! The manifold might not have a global chart. It's not a chart for the tangent space, it's a chart for M.
@@brightsideofmaths thanks for the reply ! no what I meant is could we "forget" the manifold M and just consider the tangent space TpM, which is already a vector space, to define a basis without using charts from the atlas of M. But I realized that we need this construction of the coordinate basis if we want to do meaningful calculations on the manifold anyway.
If we forget the manifold, then the tangent space is not interesting anymore ;)@@StratosFair
How to watch the hidden videos in Manifold series? What is the minimum cost to watch that series? Kindly do reply sir .
Minimum amount is 8 EUR a month for early access: steadyhq.com/en/plans/c61e16e5-454b-4f4c-bd9e-7ac46093d056
do i have to subscribe to watch part 23-27?
Yes! As an Early Supporter, you can watch all videos before :)
@@brightsideofmathsaber auch ohne Abo, kann ich die Videos in ein paar Wochen/Monaten auf UA-cam sehen ?
Thank you! :D