Manifolds 14 | Submanifolds

[@lexinwonderland5741]

FANTASTIC!! It always makes my day to see this series get updated, thanks professor! Have a great day!

[@tensorfeld295]

Jippie! A new manifold-video! :D Thank you!

[@pacificll8762]

Thank you for your videos !

[@Hold_it]

Thanks a lot!

[@redaabakhti768]

Keep up the great work thanks so much

[@PunmasterSTP]

Submanifolds? More like "Super information that's always gold!" 👍

[@tpvdwc]

I guess the submanifold is defined by using the subset topoogy? Or: Manifold (M,T) has manifold (N,S) as submanifold if N \subseteq M and (N,S) is the subset-topology of (M,T).

[@brightsideofmaths]

A submanifold is more than just a topological space. The charts are important! This is the essential ingredient here in the definition. Sometimes, one only introduces submanifolds of R^n as the main object (and does not talk about abstract manifolds). It turns out that this approach can also cover the whole theory.

[@tensorfeld295]

I think we have a regular submanifold with subspace topology (also defined via embeddings, also called embedded submanifolds). Immersed submanifolds with other topologies inhereted from the immersion are a different story ... but i am not an expert. Let M,N be manifolds and f: N -> M a one-to-one immersion then f(N) can have the subspace topology or the topology inhereted from f .... Theorem: If f is an embedding, then f(N) is a regular submanifold.

[@brightsideofmaths]

@@tensorfeld295 Yes, with embeddings you can do it like this.

[@proexcel123]

Hello, what textbooks/books do you recommend for studying manifold theory? Are there any that gives such visual representations like yours without losing that much of rigour in the mathematics? Pls keep on this series also, I love it a lot!

[@brightsideofmaths]

I like the books by Lee. However, I don't know any with a lot of visualisations.

[@proexcel123]

@@brightsideofmaths As in John Lee's 'Introduction to Smooth Manifolds'? Or Jeffrey Lee's "Manifolds and Differential Geometry"?

[@brightsideofmaths]

@@proexcel123 John Lee's book :)

[@proexcel123]

@@brightsideofmaths Oh thanks I will go look at it! Looking forward to your next videos for this series :)

[@michaelschnell5633]

Hi Bright ! I got a rather in-depth question (or set of questions), that obviously will be interesting with this series of videos (even though initially motivated by trying to understand ART). But in fact at the moment I supposedly should not come up with that, as they are deeply related to the paradigm of curvature, which has not been covered here yet (as we first must digest multi variable calculus for that). Is there some other place something like that might be discussed, of should I just wait until curvature will be introduced here ? -Michael

[@brightsideofmaths]

You can ask any question here. We will discuss curvatures in this series for sure!

[@StratosFair]

R^n with the standard smooth structure only has the identity as a chart, right ? So if we want to consider submanifolds of R^n (like open balls for instance) we would need to add more charts to the original atlas

[@brightsideofmaths]

As always: for smooth manifolds we go to a maximal atlas. So all charts are already in.

[@paperstars9078]

have you considered adding the classic "banküberweisung" for people that might want to support you this way? There might be a subset of people watching your channel that don't have paypal, but still want to support you.

[@brightsideofmaths]

Yes, this is an option since I started the channel. Maybe it's not directly findable.

[@brightsideofmaths]

Here are all the options: thebrightsideofmathematics.com/support/

[@paperstars9078]

@@brightsideofmaths ah, thank you

[@brightsideofmaths]

@@paperstars9078 I thank you :)

[@axog9776]

will you make a homology theory series one day

[@brightsideofmaths]

Yes, some day :)