Manifolds Part 2.wmv

Here we describe what might be called the Macroscopic structure of Manifolds. Includes a detailed diagram of the features of M. Construction of Charts aka Coordinate Patches on M. The atlas of charts and the C-infinity manifold condition on the common boundaries of adjacent charts. Construction of a continuous bijective map between Charts of M and the Cartesian product space of real numbers, R^n. Sub-Manifolds. Construction of a Linear Space of points in R^n. Epsilon neighborhoods in R^n as linear spaces of differentials. Construction of differentials on M.

[bundlemaker]

Thank you for the clarification. Sometimes I manage to over-complicate even the most sophisticated subject:) This clears it up. I'm getting this stuff slowly but surely and it's a blast for me, being 51 and never having had a chance at college the first time around!

[rasraster]

Thank you for this video. Your teaching is very clear. 

I have one question. On slide 15 you note that the condition of continuity on F requires that open sets in the range correspond to open sets in the image. (And I assume it's true the other way as well since F is invertible.)

If I understand correctly, topologically speaking open sets on R can be a union of disjoint open intervals. Your analysis concludes that the x_k lies in an interval a_k < x_k < b_k. Does this follow, given that the open set on the corresponding dimension of R_n might consist of disjoint open intervals?  

Thanks very much.

[Mathview]

Ah... Ok, interesting Q: For a function to be a coordinate function, we require it to have some pretty strong properties. Here's a laundry list of properties for a good coordinate function. In the case of C-infinity manifolds, F is required to be C-infinity differentiable, and continuously invertible in the sense of the inverse function theorem, and to have these properties throughout its chart. Further if F is defined on a chart, it needs to obey boundary conditions at the chart boundary.

[bundlemaker]

I'm a bit lost here. By coordinate function, do you mean a function of the coordinate variables for the particular chart? ( for example some curve represented as a function of the coordinates specific to the patch) Or is a coordinate function the thing that describes the coordinates themselves?

[albertkalden]

Hi Is the neighbor of a point in the map of Manifold a collection of points?

[Mathview]

Ah...Ok. A manifold consists of points of a topological space. Further, the coordinates of each point p in the manifold M are given by a bijective coordinate function F:M to R^n where n is the dimension of M. E.G.points of the 3d Euclidean manifold E^3 can be identified by F(p) = (xp, yp, zp) giving the Cartesian coordinates of each point p in E^3 (w. Euclidean metric.) Roughly, elements of the topology of E^3 can be described by a base of epsilon neighborhoods, and their arbitrary unions.

[ndrombes]

On Slide 15, you say that F is continuous, and so it maps open sets in M to open sets in R^n. Doesn't F need to be open, instead of continuous (i.e. the inverse map F^{-1} is continuous) for this to be true?

[ndrombes]

Which it is in this case, of course, since F is a bijection, but even so.