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What Is a Manifold?

bastian.rieck.me

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Re: What Is a Manifold?

#21

Post by mlevental »

lordnacho wrote: post #8

Is the conclusion that a manifold is a space that looks like a lower dimensional space that it really is?

no. there's no requirement for lower dimension. like another person that responded to you R^3 is 3 dimensional manifold. a (smooth) manifold is a space you can do calculus on. that means you have a consistent notion of distance (so that you can figure out when points are close together, ie you approach limits). the set of charts and continuous transformations between them is what encodes this constraint.

Re: What Is a Manifold?

#22

Post by Topolomancer »

JulianWasTaken wrote: post #11
termiefoo wrote: post #2

what ain't a manifold at this point

Besides the other answers, things also start to get interesting when you add some qualifiers. E.g., asking for a "differentiable manifold" now asks for manifolds with enough additional structure to do calculus on top of.

Good point; I originally wanted to talk about the Poincaré conjecture as well, but then I realized that this would make the post even longer. Do you have some ideas about other interesting topics?

Re: What Is a Manifold?

#23

Post by JulianWasTaken »

Topolomancer wrote: post #22
JulianWasTaken wrote: post #11

Earlier quoted context omitted.

Besides the other answers, things also start to get interesting when you add some qualifiers. E.g., asking for a "differentiable manifold" now asks for manifolds with enough additional structure to do calculus on top of.

Good point; I originally wanted to talk about the Poincaré conjecture as well, but then I realized that this would make the post even longer. Do you have some ideas about other interesting topics?

To be honest it's not even my field so I know a lot less than I wish I did but yeah agreed it has no shortage of interesting things to talk about :)

Poincaré Conjecture is yeah interesting to explain though to be honest I vaguely recall an article in similar style to your post that explained it.

Possibly something like describing gradient descent on a manifold is interesting to this audience? Or maybe a post on Flatland? Many possibilities really on good follow ups.

Re: What Is a Manifold?

#25

Post by joncrane »

It's funny, because in my much younger days I dropped out of college and worked as a mechanic for 3 years. In the automotive world and internal combustion World, manifold has a much different meaning, though related.

Basically, a manifold means something that takes the flow of gases from a one-to-many or a many-to-one.

An intake manifold takes one single entry point for air feeding the engine and splits up into a separate input for each cylinder.

An exhaust manifold takes the hot exhaust from each cylinder separately and combines them into one big pipe.

Re: What Is a Manifold?

#26

Post by hansen »

math_and_stuff wrote: post #24
username90 wrote: post #4

Earlier quoted context omitted.

Things with edges, like disks and finite lines.

Both of those are typically referred to as manifolds with boundary. The setting of Stokes Theorem is often manifolds with boundary.

Mathematicians messed up here… A manifold with boundary is not a manifold. But a manifold is a manifold with boundary (the empty set).

Re: What Is a Manifold?

#28

Post by xelxebar »

If anyone has questions (especially technical ones), the ##math channel on freenode is pretty phenomenal. It's quite active, with quite a few grad students, across a range of fields. The best part is how friendly and helpful they are to learners.

Anyway, highly recommended.

Re: What Is a Manifold?

#29

Post by srean »

man-and-laptop wrote: post #10
srean wrote: post #6

Earlier quoted context omitted.

A fractal isnt a manifold. Discrete sets are an edge case. They are considered to be 0 dimensional manifolds to make them fit, but there is a not much that manifold theory has to say about them.

> A fractal isnt a manifold Some are. A Koch curve for instance.

I believe you are correct. Koch curve is a continuous curve so its topological dimension would be 1 (not its fractal dimension).

Re: What Is a Manifold?

#30

Post by xelxebar »

_v7gu wrote: post #12

All is well until the poor bug finds out it's on a Klein Bottle, or worse yet RP2. Is there any way to distinguish between orientable or nonorientable surfaces when you're simply walking on it? If the bug was also 2d, I might have had some ideas hut I'm totally blanking for 3d.

Are you familiar with the construction of a torus by taking a square and "gluing the edges together"? It's effectively the game Astroids, where going off one edge of the screen wraps around to the opposite edge.

If, instead, you twist the edges 180 degrees before gluing, we then get the Klein Bottle. I.e. when you warp from, say, the left edge to the right edge, you also get flipped vertically.

I'm guessing you're already familiar with that, but by a similar construction, we can start with a (hyper-)cube and glue faces together to get higher dimensional analogues.

To my knoweldge the easiest visualization of a non-orientable 3D manifold is just by thinking of a cube that warps around to the opposite face. However, in addition, it also reflects you into your mirror image. This is essentially a Klein bottle in one higher dimension.

It's then not too hard to start grappling with the 4D case and beyond!

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