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What is a manifold?

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Re: What is a manifold?

#61

Post by perihelions »

WhyOhWhyQ wrote: post #49
tamnd wrote: post #48

I first learned about manifolds through Introduction to Smooth Manifolds by John M. Lee. The book is dense but beautifully structured, guiding you from basic topology to smooth maps and tangent spaces with clear logic. It demands focus, yet every definition builds toward a deeper picture of how geometry works beneath the surface. Highly recommended.

It's truly the best book on Smooth Manifolds, though if you'd like a gentler approach which is still useful, then I suggest Loring Tu's books. Lee's Topological Manifolds book is also very nice. His newest edition of the Riemannian manifolds book requires selective reading or it'll slow you down.

What's the relation between the different Lee manifolds? Is it a sequence you're supposed to read in order?

Re: What is a manifold?

#63

Post by mmooss »

brosco wrote: post #59
mmooss wrote: post #50

I rarely see manifolds applied directly to cartographic map projections, which I've read about a bit, though the latter seem like just one instance of the former. Does anyone know why cartographers don't use manifolds, or mathematicians don't apply them to cartography? (Have I just overlooked it?)

One reason is that it would be like hanging a picture using a sledgehammer. If you're just studying various ways of unwrapping a sphere, the (very deep) theory of manifolds is not necessary. I'm not a cartographer but I would assume they care mostly about how space is distorted in the projection, and have developed appropriate ways of dealing with that already. Another is that when working with manifolds, you usually…

Thanks. I've thought about those possibilites, but I really don't know the reasons.

> On a sphere or circle, you can get an "almost global" coordinate system by removing the line or point where the coordinates would be ambiguous.

Applying cartography to manifolds: Meridians and parallels form a non-ambiguous global coordinate system on a sphere. It's an irregular system because distance between meridians varies with distance from the poles (i.e., the distance is much greater at the equator than the poles), but there is a unique coordinate for every point on the sphere.

Re: What is a manifold?

#64

Post by seanhunter »

lisper wrote: post #39
elashri wrote: post #6

This reminds me of how physicists will define a tensor. So a second rank tensor is the object that transforms according as second rank tensor when the basis (or coordinates) changes. You might find it circular reasoning but it is not, This transformation property is what distinguishes tensors (of any rank) from mere arrays of numbers. Looking at things from abstract view does allow us not to worry about how we visual…

> You might find it circular reasoning but it is not Um, yes it is. "A foo is an object that transforms as a foo" is a circular definition because it refers to the thing being defined in the definition. That is what "circular definition" means .

To be fair to physicists, the standard physicists' definition isn't "a tensor is a thing that transforms like a tensor", it's "a tensor is a mathematical object that transforms in the following way ".

When people say "a tensor is a thing that transforms like a tensor" they're using a convenient shorthand for the bit that I put in angle brackets above.

My favourite explanation is that "Tensors are the facts of the universe" which comes from Lillian Lieber, and is a reference to the idea that the reality of the tensor (eg the stress in a steel beam or something) is independent of the coordinate system chosen by the observer. The transformation characteristic means that no matter how you choose your coordinates, the bases of the tensor will transform such that it "means" the same thing in your new coordinates as it did in the old ones, which is pretty nifty.

https://www.youtube.com/watch?v=f5liqUk0ZTw&pp=ygURdGVuc29yc...

Re: What is a manifold?

#65

Post by mejutoco »

aleph_minus_one wrote: post #55
mulmen wrote: post #54

Earlier quoted context omitted.

Seems superficial. If a simple concept is presented in a complex way what did you actually learn?

Often, if the concept is presented in a more complex way the reason is that the author wants to emphasize and explain how the concept relates in a non-trivial way to some other deep concept; thus you learn a lot more than when the author explains things in the most simple (and shallow) way.

IMO the most common reason why something is presented in a more complex way is that it is badly explained.

Of course, most common or not, each case is different.

Re: What is a manifold?

#66

Post by mathgradthrow »

What a terrible article. Can anyone who is not a mathematician tell me one thing they learned from this?

The naked term "manifold" in its modern usage, refers to a topological manifold, loosely a locally euclidean hausdorff topological space, which has no geometry intrinsic to it at all. The hyperbolic plane and the euclidean plane are different geometries you can put on the same topological manifold, and even does not depend on the smooth structure. In order to add a geometry to such a thing, you must actually add a geometry to it, and there are many inequivalent ways to do this systematically, none of which work for all topological manifolds.

Re: What is a manifold?

#67

Post by tamnd »

moralestapia wrote: post #58
tamnd wrote: post #51

Earlier quoted context omitted.

I learned about Calabi Yau manifolds a long time ago and have forgotten most of the details, but I still remember how hard the topic felt. A Calabi Yau manifold is a special kind of geometric space that is smooth curved and very symmetrical. You can think of it as a shape that looks flat when you zoom in close but can twist and fold in complex ways when you look at the whole thing. What makes Calabi Yau manifolds spe…

>their curvature balances out perfectly so the space does not stretch or shrink overall Could you elaborate a bit on this? I find it fascinating. Thanks. >The shape of a Calabi Yau manifold affects how particles and forces behave [...] Do you know if there's any experimental evidence of this?

> Do you know if there's any experimental evidence of this?

As my knowledge, there is no direct evidence that Calabi Yau manifolds describe real extra dimensions. In string theory, these shapes are used because they fit the math and preserve symmetries like supersymmetry. Experiments have not found signs of extra dimensions or supersymmetric particles, so Calabi Yau manifolds remain a beautiful theoretical idea, not something confirmed by observation.

Re: What is a manifold?

#68

Post by tamnd »

moralestapia wrote: post #58
tamnd wrote: post #51

Earlier quoted context omitted.

I learned about Calabi Yau manifolds a long time ago and have forgotten most of the details, but I still remember how hard the topic felt. A Calabi Yau manifold is a special kind of geometric space that is smooth curved and very symmetrical. You can think of it as a shape that looks flat when you zoom in close but can twist and fold in complex ways when you look at the whole thing. What makes Calabi Yau manifolds spe…

>their curvature balances out perfectly so the space does not stretch or shrink overall Could you elaborate a bit on this? I find it fascinating. Thanks. >The shape of a Calabi Yau manifold affects how particles and forces behave [...] Do you know if there's any experimental evidence of this?

> Could you elaborate a bit on this?

Please correct me if I am wrong, I have not touched this subject in a long time and only have some intuition. Here is how I understand it:

A manifold is a kind of space that looks flat when you zoom in close enough. The surface of a sphere or a doughnut is a 2D manifold, and the space we live in is a 3D manifold. A Calabi Yau is one of these spaces but with more dimensions and extra symmetry that makes it very special.

In geometry there are several ways to describe curvature. The most complete one is the Riemann curvature tensor, which contains all the information about how space bends. If you take a specific kind of average of that, you get the Ricci curvature tensor. Ricci curvature tells you how the size of small regions in space changes compared to what would happen in flat space.

Imagine a tiny ball floating in this curved space. If the Ricci curvature is positive, nearby paths tend to come together and the ball’s volume becomes smaller than it would in flat space. If the Ricci curvature is negative, nearby paths move apart and the ball’s volume grows larger. If the Ricci curvature is zero, the ball keeps the same volume overall. So when I said “the space does not stretch or shrink overall” I was describing this situation: the Ricci curvature is zero, which means the space does not expand or contract on average compared to flat space.

The space can still have complicated twists and bends. Ricci curvature only measures a certain type of curvature related to volume change. Even if the Ricci tensor is zero, there can still be other kinds of curvature present. The curvature balances out is just an intuitive way to express that the volume effects cancel when you take the average that defines Ricci curvature. It does not mean the space has matching regions of positive and negative curvature in a literal sense, but rather that the mathematical combination producing Ricci curvature sums to zero.

Noe back to definition: A Calabi-Yau manifold is defined as compact (finite in size), complex), and Kähler (it has a compatible geometric and complex structure), with a first Chern class equal to zero. Yau’s theorem proves that such a space always has a way to measure distances so that its Ricci curvature is exactly zero. So when I said “the curvature balances out perfectly so the space does not stretch or shrink overall” I meant it as an intuitive description of this Ricci flat property. The space is not flat like a sheet of paper, but its internal geometry is perfectly balanced in the sense that there is no net expansion or contraction of space.

Re: What is a manifold?

#69

Post by senderista »

mmooss wrote: post #63
brosco wrote: post #59

Earlier quoted context omitted.

One reason is that it would be like hanging a picture using a sledgehammer. If you're just studying various ways of unwrapping a sphere, the (very deep) theory of manifolds is not necessary. I'm not a cartographer but I would assume they care mostly about how space is distorted in the projection, and have developed appropriate ways of dealing with that already. Another is that when working with manifolds, you usually…

Thanks. I've thought about those possibilites, but I really don't know the reasons. > On a sphere or circle, you can get an "almost global" coordinate system by removing the line or point where the coordinates would be ambiguous. Applying cartography to manifolds: Meridians and parallels form a non-ambiguous global coordinate system on a sphere. It's an irregular system because distance between meridians varies with…

The problem is that this global coordinate system isn't a continuous mapping (see the discontinuity of both angular coordinates between 2*pi and 0). Manifolds are required to have an "atlas"[0]: a collection of coordinate systems ("charts") that cover the space and are continuous mappings from open subsets of the underlying topological space to open subsets of Euclidean space, with the overlaps between charts inducing smooth (i.e., infinitely differentiable) mappings in Euclidean space.

Colloquially, this means a manifold is just "a bunch of patches of n-dimensional Euclidean space, smoothly sewn together."

A sphere requires at least two charts for an admissible atlas (say two hemispheres overlapping slightly at the equator, or six hemispheres with no overlaps), otherwise you get discontinuities.

[0] https://en.wikipedia.org/wiki/Atlas_(topology)

Re: What is a manifold?

#70

Post by robotresearcher »

getnormality wrote: post #21
robot-wrangler wrote: post #12

Earlier quoted context omitted.

I'm always surprised more people don't know about Quanta. Seems like it's currently the best science journalism out there, and IMO a very strong candidate for the single best place on the internet that's not crowd-sourced. The mixture of original art and technical diagrams is outstanding. Podcast is pretty good too, but I do wish they'd expand it to have someone with a good voice reading all the articles. Besides not…

It's because of their Simons Foundation support, but not only because of that. I mean, I invite anyone to name another billionaire pet project of comparable quality.

Carnegie Libraries, Nobel Prizes, Rhodes scholarships?
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