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What Is a Manifold?

bastian.rieck.me

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Re: What Is a Manifold?

#81

Post by mindcrime »

joncrane wrote: post #25

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 sep…

A similar notion exists in the fire service, where a manifold is an appliance for combining or splitting water supply lines. A typical example of a fire service manifold would be an appliance with 3 2.5" threaded ports and a single 5" Storz port, which can be used to either merge multiple 2.5" or 3" supply lines into one 5" supply line, or to divide the flow from a 5" line up into multiple smaller lines.

Example: http://fireflowtechnology.com/wp-content/uploads/2015/04/100...

Re: What Is a Manifold?

#82

Post by hhmc »

mlevental wrote: post #35
xelxebar wrote: post #28

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.

this in general accurate but I would warn people to have a thick skin in there because there are some seriously toxic people in there (TRW3W or whatever his/her name that was fairly knowledgeable but hung out in there I think purely to assert his/her superiority).

I still recognize that name/handle and I haven't logged into ##math for ~10 years, so I can certainly corroborate what you're saying.

Re: What Is a Manifold?

#83

Post by chadcmulligan »

amai wrote: post #67

So a manifold is a surface?

it can be, I think of a manifold (probably naively) as the mathematical abstraction that you end up with if you start from Euclidean space and fiddle with different features - e.g. if you're in a Euclidean space how do you get from one point to another - what are the minimum requirements? What if we want calculus to work what are the minimum requirements? If we're on a surface and we walk around is there any way to tell what shape that surface is?

Re: What Is a Manifold?

#84

Post by improbable22 »

arduinomancer wrote: post #76

I think the natural next question is how does a bug in R3 check if its part of a 4-dimensional manifold? This is related to people measuring the curvature of the universe right?

The article squeezes together two notions, of whether a manifold is curved, and whether it's a submanifold of some higher-dimensional one.

Sub-spaces of R^3 are a useful way of generating and picturing examples of 2D manifolds. But these things exist by themselves. As the article mentions with angles of a triangle, you can tell that a 2D manifold is curved from living inside. Likewise you can tell whether a 3D manifold is curved of not, without any mention of a 4th dimension.

Questions of curvature of the universe are a step harder, as we are talking about 3+1-dimensional space-time. But the slice of constant time is a 3D manifold, and as far as we can tell right now, it appears to be flat.

Re: What Is a Manifold?

#85

Post by adrianratnapala »

joncrane wrote: post #25

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 sep…

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

A perfectly sensible meaning given the construction of the word. There is something about mathematics and linguistics (and to some extent CS) that encourages the creation of confusing, meaningless names like "accusative (case)", "(algebraic) ideal" and "(geometric) manifold".

Re: What Is a Manifold?

#86

Post by jazzyjackson »

TeMPOraL wrote: post #58
joncrane wrote: post #25

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 sep…

Isn't this like a multiplexer/demultiplexer in electronics/signal processing?

You could at least say that the ground plane on a circuit board is an exhaust manifold

Re: What Is a Manifold?

#87

Post by alcolade »

improbable22 wrote: post #80
alcolade wrote: post #77

If the bug were truly mathematical, he would notice the curvature of S^1 and S^2 without walking around (since curvature is defined at each point), but that is obviously very pedantic of me and shows why I wouldn't write an article nearly as interesting as this one.

But S^1 isn't curved. At least if we mean intrinsic curvature, which is the kind that belongs to the manifold itself. S^2 is curved, but locally you can't tell it's not (say) RP^2.

I was thinking of the Riemann curvature tensor which is just some measure of curvature at each point. If the bug were at some point of S^2 he would notice the curvature.

Of course, this is assuming S^2 is getting its geometry from a certain embedding into R^3 that comes to mind. You could define different Riemann curvature tensors over S^2 that may have zero curvature in certain places, like squishing a balloon against a flat table for example.

I guess my point was that topology and curvature are different things (like you pointed out with your comment about RP^2), and saying that "a sphere looks flat locally" is missing the point!

Also, it didn't occur to me that this notion of curvature doesn't make sense in 1d, i.e. for S^1 like you said.

Re: What Is a Manifold?

#88

Post by tgb »

improbable22 wrote: post #79
tgb wrote: post #70

Earlier quoted context omitted.

Yes, I took "circle" to mean any closed path since the difference is immaterial in generic manifold. You wouldn't normally be talking about "orienting a circle" in the manner the original poster did, but I see now what they mean. I took it to mean "choose an orientation along a closed path" which would be impossible to do for some (but not all) closed paths in a non-orientable surface.

A closed path is S^1, which is certainly orientable, no matter what it's embedded into. You just draw an arrow on the line. ColinWright's point is that drawing such an arrow on a tiny circle is equivalent to drawing the letter R on the 2D manifold. It gives a local orientation to the 2D surface. (If you wish you may think of this as an arrow into some 3D embedding space, but you don't have to.) If the 2D space is ori…

Yes, you can always choose an orientation of the path but not necessarily of the manifold along the path. I understand what he was saying now and misunderstood since it is not the usual way one talks about specifying an orientation at a point and in fact had imagined he was specifically specifying a "big" circle, like a nontrivial loop on the torus. You'd more typically talk about choosing a basis of the tangent space at a point - presumably ColinWright wanted to avoid involving extra definitions like tangent space and so chose a visual definition that wasn't quite as precise. I don't think there's any further confusion.

Re: What Is a Manifold?

#90

Post by improbable22 »

alcolade wrote: post #87
improbable22 wrote: post #80

Earlier quoted context omitted.

But S^1 isn't curved. At least if we mean intrinsic curvature, which is the kind that belongs to the manifold itself. S^2 is curved, but locally you can't tell it's not (say) RP^2.

I was thinking of the Riemann curvature tensor which is just some measure of curvature at each point. If the bug were at some point of S^2 he would notice the curvature. Of course, this is assuming S^2 is getting its geometry from a certain embedding into R^3 that comes to mind. You could define different Riemann curvature tensors over S^2 that may have zero curvature in certain places, like squishing a balloon again…

Not only is there no intrinsic curvature in 1D, but the 2D and 3D versions are also simplified special cases.

In 2D there is only scalar curvature, i.e. you don't need the whole Riemann tensor, just one number (at a given point).

In 3D you need the Ricci tensor, but still not the full curvature tensor. This is still much simpler, for instance IIRC this is why you cannot have gravitational waves in 3 (meaning 2+1) dimensional spacetime. In 4D you get the full complication.

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