Manifolds Part 1.wmv

What are Manifolds? Qualitative walk through of the definition of manifolds as topological spaces having differential structure. Coordinate functions mapping points of M to Cartesian products of the real numbers. Coordinate charts or patches and the atlas of a manifold. Coordinate functions of the atlas form a bijective map between M and R^n. Manifolds have macroscopic and microscopic structures.

Category:

Education

Tags:

• Mathematics
• Manifold
• Differential
• Space
• Structure
• Riemann
• Coordinates
• Linear
• Tensors
• Vectors
• Chart
• Patch
• Atlas
• Topological
• Topology

[eyalenjoi]

As I Understand, if I got a coordinate function F:B->R^n then I can define a function to R^m m>n. so dim(M) is the minimal value?

[Mathview]

@eyalenjoi @eyalenjoi Ah...yes. Good question. A manifold M consists of a space of points (a topological space) and a mapping (system of coordinates) such that each point corresponds to a UNIQUE point in R^n where n = Dim(M). The coordinate function is bijective between points in M and coordinates in R^n. Bijection implies a unique value of n, the dimension. It's possible to embed M in a higher dimensional manifold Q with d>n. M becomes a sub-manifold of Q. Embedding theorems are used do this.

[Mathview]

@eyalenjoi So there's some subtle aspects to your question. For example, it's not always obvious to see how to actually embed a given manifold into a higher dimensional manifold. A surprising example is the Klein Bottle. The KB is a 2d manifold that has a simple embedding in R^4, but not so in R^3. We can make a model of a Klein Bottle in the glass shop, but the surface intersects itself, i.e. double valued in R^3. So the surface in R^3 is an immersion, but not an embedding.

[MrAdammclean]

Are the coordinate generating functions related to sheaves of continuous functions on a topological space ? Are sheaves a further abstraction of the ideas you are presenting here ?

[Mathview]

@MrAdammclean Ok, interesting question. Manifolds must have coordinates to be officially manifolds. So we have the topological space of points, their coordinates in R^n, and we must have a bijective map between points of M (dim=n) and R^n. Call it a coordinate (generating) function F:M -> R^n. All the above constitutes the manifold M. We have the option to attach other stuff directly to open sets in the topological space M, which could be done via sheaves...is a partial answer. Great question.

[Mathview]

Here we will not focus on the topological aspects of manifolds. It is important to simply recognize that the sets of points in manifolds have certain "technical" properties that allow sets of points in M to be well behaved with no funny or pathological behavior. One might say, "We take it as given that the manifolds we discuss will have sufficient topological properties as needed to support the constructions used."

[poikonen01]

Thank you so much for this video--very good introduction.

[omicron8251]

Sorry, but what is the meaning of the picture that appeared in the beginning? System of how vectors operate on manifolds?

[MrAdammclean]

Perhaps you might be able to address the nature of sheaves in some future presentation. The idea of a sheaf is so abstract that I find it difficult to get some intuitive grasp of it. I would benefit from your particular approach with your use of diagrams and a thorough explanation of the formalism.

[eedahl]

I very much like how you --- at the end there --- emphasize the sheer volume of structure associated to a manifold. Pretty much like your discussion on the metric tensor. There seems to be no end to the properties of manifolds :-D

[TheMathKing]

Amazing discussion on these topics! It's been a while since I've heard such refreshingly rigorous mathematics, since I've been doing lots of physics. Do you have a Ph.D in mathematics?