What is a manifold?

A visual explanation and definition of manifolds are given. This includes motivations for topology, Hausdorffness and second-countability.

If you want to learn more, check out one of these (or any other basic differential geometry or topology book):

- M. Spivak: "A Comprehensive Introduction to Differential Geometry"
- M. Nakahara: "Geometry, Topology and Physics"
- J. W. Milnor: "Topology from the differentiable viewpoint"

[suirahplanogemo]

This video definitely is not "noob-friendly".

[evanm]

The mathematical definition of a manifold: A subset of R^n locally diffeomorphic at all points to a neighbourhood of the origin in R^m, where m<n. The engineering definition: Something that bends.

[thebiber]

I'm a physicist and teacher and I took a differential geometry course 6 years ago. Nowadays I'm repeating everything I've learned and I slowly grasp what's really important for me to understand manifolds. This video suffers from the same problem every mathematical textook has for this topic. You define a manifold by several terms, highlighting smoothness above everything else as if it's the most important thing but in the end I have no clue how to do operations on a concrete example of a manifold.

The most important terms for physicists are the Riemann metric, the Levi-Civita-connection and the Riemann curvature tensor and for that you don't really need to know what Hausdorff, homeomorphisms, smoothness and all the other abstract terms really mean. A good introduction of manifolds should use concrete examples like a sphere, a torus or a cylinder and explain those terms for these specific examples.

Another problem I face with mathematical introductions is lack of motivation of the definitions. Most mathematicians just define a concept, state a theorem and prove them but they don't explain why a concept is important or why a generalisation is necessary. Before defining a manifold you first have to learn that it is a generalisation of a submanifold of Euclidean space (curve, surface, hypersurface, etc.). There you have to learn what a tangent space is, what curvature means and the theorema egregium, which states that curvature can be measured without referring to the surrounding space. Only then you have the motivation to understand that a manifold is the generalisation of submanifolds by ignoring the surrounding space completely.

This video ignores all these important aspects and does just what every mathematics textbook do, which is bad teaching.

[piaopiaokeke]

I have a math degree. This made sense to me. But honestly, I don't think this would help a non-PhD level phycisist

[Anonymous-lwzy]

If the point of this was to take someone with a good math background but almost no differential geometry and give them a fast introduction to the key concepts and terminology so they can understand other material dense with differential topology terminology (e.g. the SageManifold software documentation), then this was perfect. That is what I was looking for and this superbly met that need. My background is electrical engineering undergrad and some theoretical physics. It took a few viewings, but so what - it is only 3:50 long. Amazingly great job taking a very complex topic and lots of terminology and reducing it down to a clear, short video. Thanks for doing it!

[AbdoEldesokey]

Man, this made it even more complicated!

[pascalemp]

Coming from somebody with a degree in Mathematics, I never fully understood the notion of manifolds when studying Topology, but this video (in particular the introduction) elaborated on concepts I've been confused about for years; and within a matter of minutes and it made complete sense! Thank you very much!

[PomeraniansRock]

I have a short presentation on manifolds due tomorrow for my Differential Equations class. This video saved my life! Thanks.

[RomanZillek]

@0:42 with the map <> space table there is a typo|error in with the "linear<>vector space" line: its v*f(y) not y*f(y).

[acatisfinetoo]

The way you convey these complex subjects is just beautiful. This is the way mathematics SHOULD be taught

[fbkintanar]

Nice video. To the list of references at 3:35, you might want to add Chris J. Isham "Modern Differential Geometry for Physicists". He takes a modern coordinate-free approach from the start, and I found his chapter introducing topology helpful in developing intuitions. He also mentions some less studied ideas from topology, including the lattice structure which I found to be insightful and related to certain notions from computer science (locale theory, discussed in Steven Vickers "Topology via Logic")

[migonarvo]

undergrad here, man, I was so LOST, thank you so much for this explanation and the book recommendations, pretty cool

[LunaticBiker]

Sounds cool, maybe I'll understand it someday :))

[Simon-xitb]

I know basic maths like additon and multiplication. So how can i fix the manifold on my car ? i didnt understand anything shown here.

[DanielEstrada]

This video is very helpful! I'd love to see more.

[michaeljehlik]

I r4eally like this video, the graphics and teaching style are exceptionally good! This visual approach is easy to follow and understand. I'm looking forward to more videos like this : )

[xodarap]

at 1:09 that Hausdorff condition; is that a way of saying that the/a manifold cannot intersect with itself? There is another video [much less succinct than this one] in which it asserted that an M dimensional manifold can be embedded in in a space of 2M dimensions [but seemingly not of less than 2M]

[artforartssakerecording]

If someone could help me. I am looking for the geometric forms, spirals that grow forward and then return at the same point (and grow (if it possible ) backward), and so on? A Kind of the loops in the growth in exponential expansion.

[criskity]

I think of map projections as an application of this. The earth's surface is a spheroid, so a region of this surface is mapped via the coordinate function (the projection method: Mercator, Mollweide, Cylindrical, Azimuthal Equidistant, etc.) onto a region in R^2 (a flat map).

[deepbayes]

I don't have a PhD in physics, yet this helped. In fact, one of the best videos on the topic. I would like to see more of this material on geometry and manifolds.