What is a Manifold ? - Intuition and Definition

We discuss the idea of manifolds informally, and then give a formal definition, discussing the underlying concepts of topological spaces, continuous maps and locally Euclidean spaces. We define topological manifolds, smooth manifolds and Riemannian manifolds.

[devon]

Any thoughts on the combination of ChatGPT and the Wolfram plug-in for learning some of these concepts?

[rajehkumarmishra]

hi, richard can you suggest me a good book to study manifolds from optimisation perspective

[ohault]

I would suggest to add some counterexamples and some visual illustrations about Tangent spaces.

[thuankhuongminh]

I havnt got the time to watch your video yet!!! I will watch it latter it looks very interesting.
My view on the manifold is that it got metrical structure, group structure, and hyperbolic structure.
The manifold is complex and we do not understand it yet beyond 3-manifold(I might be wrong, very new on topology).
I think that one day if we understanded it better we will able to figure out a manifold that measure the geometry of space in the microscale.
And there well be a new subject on quantum mechanics based on deterministic rather than probrabilistic.
I got my degree in Theoretical Phyiscs in 2012, and have been an ameter mathematian since.
I am going to do my master and PHD in mathematics to study this manifold.
If you would like to colaberate with me, ( I would love to colaborate)
please let me know. I would love to learn some projective geometry from you as well.