Manifolds #1 - Introducing Manifolds

Here I begin to introduce the concept of a manifold, building on our intuition gained from studying topological spaces. I will formalise all of the terminology used in this video in future sections but for now I'll be speaking in fairly loose terms about manifolds.
We will see that the essential idea behind constructing a manifold is to take a topological space and cover it in charts. These charts are open sets which are homeomorphic to R^d and are viewed as giving coordinates to a local patch of the manifold. This can equally be viewed as saying that a manifold is a topological space which is locally homeomorphic to R^d.
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[@niceperson2]

I think someone without a masters can also understand this easily. The transition from topological spaces to manifolds was beautifully shown

[@FlareGunDebate]

I've just starting this series but so far this is the best video I've found on topology and manifolds. Thank you.

[@jadavpurstudents1365]

kzbin.info/www/bejne/kGeaq35qd5uWnLc✌

[@manaoharsam4211]

A very good introduction.

[@adityaiyengar152]

This is such a wonderful series. Thanks for making it pretty basic and accessible yet interesting so as to cover the essence of manifolds. I subbed in the first 2 minutes -- literally. Looking forward to more math content!

[@mireazma]

Compared to many other videos, you explain clearly, thanks.

[@andrewsheehy2441]

This is so good. I loved the jab at the flat earthers! It would be cool to print out series of patterns on paper and then stick them together accurately with the result that you'd end up with a 3D object.

[@smileifyoudontexist6320]

One could Not argue that the Sphere recreated by gluin. All the pages of an Atlas together would only be showing the curve Spherical shape of the Space the earth is sitting in…. Flat earth in a Curved space…. Im not saying that the earth is Fla or round ,,, But most people have absolutely made up their minds what they believe very strongly, but very few have actually done their own measurements or individual thought…. Whether or not it is round or flat, People will believe what is most socially acceptable and beneficial to their own local immediate environment

[@zwitter689]

You may want to check out info on Euleriam Spirals, if you aren't already quite familiar. For instance , see www2.eecs.berkeley.edu/Pubs/TechRpts/2008/EECS-2008-111.pdf

[@bobtanto3860]

No, that would be a waste of time for this youtuber. If people can't convince themselves through imagination that small pieces of flats can combine into a sphere, then they should try it themselves so that their power of imagination can improve.

[@petermarshall8212]

Thank you for sharing your gifted understanding … really quite wonderful …

[@arnobchatterjee773]

What an amazing introduction to point set topology!

[@sofiyavyshnya6723]

Excellent videos! I love your entire series on Manifolds. You explain things so that even I can understand :) Keep up the good work!

[@WHYBmaths]

Thank you very much glad they are helpful!

[@Towalak]

Love it, good work.

[@cannoli1363]

Great explanation and such a BRILLIANT outro!

[@johnsondick3176]

Beautiful. The concept is now clearer

[@gpas80]

Clearest explanation I’ve found, thanks.

[@seminkwak]

Amazing explanation and definitions!

[@BanMidouSan]

Thanks a lot! Saludos desde México.

[@febojarlock9469]

I was about to give up on manifolds until I saw your series! Thank you dude!

[@anonymous.youtuber]

Thank you so much for your clear explanation. 🙏🏻

[@stevemenegaz9824]

Very nicely done. Easy to understand.

[@deepaksingh3533]

Wow , u r such a wonderful teacher.. god bless you sir...

[@camerontankersley3184]

This video is amazing, thank you so much

[@xDDeeDee]

Exquisite explanation, thank you :)

[@anthonyh694]

Very clear explanation, thank you

[@fizikchy]

10:04 Türkiye, long live :)) Great video series. I also visited your github page, found some pdf files so on. You're doing a great job.

[@garythomas4936]

Excellent explanation.

[@dimzog]

Exceptional, thank you sir!

[@michaelhaag3367]

lucid explanation, thank you very much

[@valor36az]

Excellent video

[@n00bphd84]

What a great fucking series! I have been building to understand optimizations on manifolds (send resources) and this has been very useful! Thank you!

[@ameerhamza4816]

that was very helpful

[@christophera3330]

Dude - you're amazing!

[@zwitter689]

Nice job with this, I like your style of explaining a difficult subject.

[@seanriopel3132]

It can be extremely difficult. The intelligence it takes to understand this stuff puts people on a different level than your average person where basic communication becomes difficult. People like this remind me of Feynman. Absolutely brilliant but it was something he had to work very hard at and never lost his ability to explain things to us mere mortals.

[@zwitter689]

@@seanriopel3132 Wholeheartedly agree with you, Sean. In addition, it takes a good level of intelligence to appreciate the difficulty you referenced and I am stronglyinclined to think that's something you have and/or exceed.

[@bobtanto3860]

I have physics background and found this to be greatly helpful. Thanks!

[@gansx8390]

Very intuitive! TBH I didn't get it at first. I checked some videos about topology then came back and I suddenly understand all these stuff.

[@rishabhgoenka9686]

Thank you so much, amazing video

[@marinajacobo3550]

This is awesome. Thank yo so much.

[@abhishekawasthi8763]

Nicely explained 👌

[@csirnetiitjammathematics7888]

Amazing pls upload more videos on manifolds

[@loden5677]

This really helped me gain an insight into elementary topology, thank you!!! If you have any book / course recommendations (or even vids/reading) please let me know :)))

[@mathscholor]

Love you Bro!

[@slash8266]

So am I right to think of manifolds as spaces that can be represented by a collection of lower dimensional sets? We can talk about the points on a sphere using a coordinate system. But there is an advantage in "dimensionally reducing it" to a set of simpler portions/patches?

[@kanefoster8780]

Great vid, ta lad

[@YazdaniBah]

Pretty good easy and understandable

[@Henry-yr2hn]

nice work

[@DrTWG]

Thanks for this explanation . Not sure about the septal manifold though.

[@igboukwu]

Thank you so much.

[@ckarcher4504]

thank you very much!

[@davebowman1968]

Thank you so much!!!!!

[@alijoueizadeh2896]

Thank you!

[@saadijalal3877]

Excellent

[@vimwik]

Thankyou for clear demonstration of what a manifold is.

[@user-wi1rj4iw9y]

Thank you! 谢谢！

[@mateusmachadofotografia8554]

Nice!

[@onlineclasses6544]

Really good from India

[@cosmophilosophyandscience1674]

Thanks sir, from india 😊

[@mukilraj9855]

Lovely

[@elir7184]

So, would it be accurate to say that a manifold is simply any topological "shape", of any dimension, that can be mapped onto a 2d chart? Or otherwise rendered into a congruent system of coordinates? Does a manifold become a manifold only when it is mapped? Also, Can a space itself, without a "shape" within it be considered a manifold?

[@WHYBmaths]

Essentially yes, although be careful since topological spaces have no notion of `shape' - this is extra geometrical structure that is only be defined once a topological space has been realised as a manifold, and then that manifold given a particular geometry. For example, the sphere S^2 is abstract and topological, it can be mapped for example using spherical coordinates, but this still has no notion of `shape' other than our pre-conceived notions of sphericity. The sphere can then be embedded into R^3 given a `shape' (by defining its extrinsic curvature - see my Einstein-Cartan introduction videos) to produce for example the round sphere (or any other ellipsoid). In answer to your last question yes, because essentially all manifolds have no intrinsic shape until extra geometric structure (distances and curvature) are defined on the manifold. At this point, a manifold is just a slightly more concrete realisation of a topological space; which uses sets of numbers as coordinates to label the points of the totally abstract topological space. Also note an N-dimensional manifold would map to an N-dimensional chart. Yes a manifold only becomes a manifold when producing a set of charts (and showing they are consistent).

[@pinklady7184]

This is my first day of learning manifolds. As 3D artist self-studying maths & physics, I understand your tutorial. I have just checked out homeomorphism, homoemorphic topology, and suchlikes. I have just subscribed.

[@WHYBmaths]

Thank you! As someone who also dabbles in 3D art I'm pleased my videos can help people from this field also! I was surprised quite how much theory is involved in 3d modelling, UV vector fields, normal maps and of course topology/homotopy/homology in general, all great stuff to learn about for modelling! Glad you finding the videos helpful!

[@ChezzyKnytt]

This is very useful, thanks. Based on this, can someone please explain to me what an orbifold is? Or do I need more information before I can understand what that is?

[@TheAIEpiphany]

Earth is actually a geoid if we are trying to be rigorous :P which we clearly aren't. Thanks for the video!

[@soccerbels7947]

Actually its an flat paper

[@h.h.c466]

The distortions will carry over in the reconstructed manifold so it will be not exactly the same we began with or still?

[@QuinnYouTube]

thanks

[@phatdo7657]

I hope you are my professor

[@1991acgs]

Question: why is the sphere S2 if it is placed in R3?

[@robertemprechtinger8203]

The description of a manifold starts at #6:50.

[@johnhammer8668]

what is the data type of elements in S1 if it is not points (real numbers) ?

[@Abhijitdas8710]

Can you help me with a question...- * How can i justify a circle is intrinsically a flat manifold??

[@jaeimp]

You did a great job at giving the intuition of manifolds, but it struck me as counter-intuitive that you kept referring to the terrestrial globe as "abstract" topological space. In what sense is it 'abstract'?

[@krokenstiv8777]

what are the applications of manifolds in physics?

[@Goldslate73]

Are mammals equivalent to a donut? (I'm pointing towards the digestive system since every mammalian body practically has a long and complex hole through it.) Just a thought.

[@joefuentes2977]

Earth is by no means a sphere! Imagine how many underground subways we made in addition to the natural holes that were always around!

[@hyperbolicandivote]

Consise explanations, thanks.

[@oxhb]

for a layperson why is a torus in R2 and not R3 space?

[@aizaimran947]

cute dog haha

[@farhanrafid8584]

i see myself on 8:26 (Bangladesh)

[@renatoaraujo6108]

I never felt so humiliated

[@websurfer352]

You could make your lesson more graspable by precisely defining some terms you employ?? For instance when you refer to an abstract topological manifold, I believe you mean that since the actual surface of the globe consists of real locations on the globe and since the topological manifold which is the 2-dimensional surface of the globe consists of points which are innumerable and cannot be enumerated since they are a continuum then yes the idea of it as a topological manifold would be abstract!! But most students when faced with such a term which they are unable to define precisely in their minds get somewhat uprooted in their thought processes and from that point on remain in bewilderment as to the actual meaning of the term!! That is what happens when an imprecisely defined idea comes which you need to work on in subsequent thinking.

[@joefuentes2977]

Ha! I knew flat earthers were right!

[@Goldslate73]

3:59 Mr. Bean's potato.

[@phantasmleechidna8650]

Wherenis jack

[@loriclark505]

okay so there is a bible verse about the manifold wisdom of God,what does that mean?

[@BeMusicalHarmonica]

Why the nose ring?

[@WHYBmaths]

¯\_(ツ)_/¯

[@CannabinatedFantasy]

try to be less distracting and get a good setup

[@philcooper279]

A very good introduction.