What is a Riemannian Manifold?

poetix

we murder to dissect
There must be someone on here who knows, and can explain it clearly. (Assume I've already glanced at the Wikipedia page, and thought "oh, right, that looks like the sort of thing a kind and patient mathematician ought to be able to explain to me").

If there isn't, or they can't be arsed, I'm going to work it out for myself* and post the answer on this thread.

Political enlightenment and/or enhanced appreciation of Deleuze may or may not eventuate.

* I may, as the very gallant Oates said, be some time.
 

Mr. Tea

Shub-Niggurath, Please
Staff member
I'll get back to you when I've had a look over my old undergrad Maths for General Relativity notes.

This interest has been piqued by use of the term by Alain Badbwoy, I assume?

Edit: dur, Deleuze, obv.

OK, well I've had a quick look at my notes and also at Mathworld and a few Wikipedia entries, so I'm prepared to have a go.

A manifold is an N-dimensional space where every point can be locally mapped to R^N, where R means the real numbers and taking R "to the power of" N just means you're using N real numbers to describe a position - so that on an infinite, flat 2-dimensional plane, any point can be described completely with two coordinates, eg. x and y on the familiar Cartesian axes.

This flat plane is also called a 2-dimensional Euclidean space. Extend it to 3 dimensions, and it's a pretty good description of real, physical space in the absence of strong gravitational fields - i.e. the space we live in. You can add a time dimension to this 3-d Euclidean space to obtain a four-dimensional space-time, a.k.a. Minkowski space-time, which forms the basis for special relativity.

Anyway, in the middle of the 19th century Riemann, building on earlier work by Gauss, started to experiment with spaces in which the axioms of Euclidean geometry (such as "parallel lines converge at infinity" and "the shortest distance between two points is a straight line") were not assumed. This was all done in the spirit of purely mathematical enquiry, as it was not imagined at this point that there could be real physical applications of this work. So rather than just working in 'spaces', Riemann and others worked with 'manifolds', where Euclid's rules for 'flat' space don't (necessarily) hold, and the mapping of points from the manifold to R^N is perhaps non-trivial. An example that's simple to visualise is S^2, the surface of a sphere, where the x and y coordinates of 2-d Euclidean space are replaced by latitude and longitude, and the shortest line between two points is no longer straight but a geodesic, a section of a 'great circle'. Also, the surface of a sphere has intrinsic curvature, which is not shared by a flat plane. (Intrinsic curvature means that a flat sheet - a piece of paper, say - cannot be deformed onto the surface without 'crumpling': thus the cylindrical main part of a wine bottle does not have intrinsic curvature, but the 'shoulder' where this part meets the neck does have intrinsic curvature, as does a sphere.)

Now a differentiable manifold is one which is locally similar enough to Euclidean to allow calculus to be performed; in other words, it has no 'sharp' boundaries or discontinuities, ensuring that (in a given mapping to R^N) the mapping is single-valued at all points, and differentiable, i.e. continuous, or having a finite derivative with respect to some coordinate system. Another ingredient we need here is the metric, which is a function g(x1,x2) that gives the shortest distance between two points x1 and x2. Then a Riemannian manifold is a real differentiable manifold with a non-negative metric, which allows intuitive geometrical quantities like length, area, volume and angle to be meaningful.

The maths here is probably a bit mangled, because I learnt the basics of this stuff from a physicist's POV. Someone like Slothrop may be able to give a more rigorous and accurate description. Anyway, general relativity is phrased mathematically in terms of Lorentzian manifolds, which are classified as 'pseudo-Riemannian' because the metric can be negative. This may sound nonsensical, as it describes a distance, but remember we are talking here about 'distances' in space-time, not space.

Hope this has been of some help.
 
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poetix

we murder to dissect
Thanks, Tea, that is indeed helpful.

Let me go over part of what you've said so far, sticking with the example of the surface of a sphere as that's quite easy to visualise.

I have a street map of Northampton town centre, which opens out into a large rectangle. The "points" on the interior of that rectangle can be seen as an open subset of the Euclidian space R^2: if we measure in inches across the surface of the map, any point "inside" the map will be 0 < x < W inches across from the left (where W is the width of the map), and 0 < y < H inches up from the bottom (where H is the height).

If I had a very large globe map of the earth, to the same scale as my town centre map, I could find Northampton on that globe and paste my map over it so that the points on the map lined up with the points on the surface of the globe. In fact, for a rectangle so small (relative to the overall size of the globe), I probably wouldn't notice that I had to stretch or crumple the rectangle into an ever so slightly non-rectangular shape in order to make it "fit".

The surface of the globe is "locally" (e.g. within the parish bounds of Northampton) similar enough to a region of 2-dimensional Euclidian space that I can map everything within such a locality to an open set within a Euclidian space, and this mapping will be homeomorphic (in other words, if two points p and q are "near to" each other in Northampton, their mapped points p' and q' will be "near to" each other in R^2. We can give a topological definition of this "nearness" or being-in-the-neighbourhood-of in terms of membership of open sets, in which case what matters is that the mapping preserves the structure of the lattice of open sets "localising" each point).

However, "globally" the surface of the globe can't be mapped in this way - no matter how you stretch it, you can't completely completely cover the surface of a sphere with a rectangular "rubber sheet" (or not without "sewing it up" at one point. Interestingly, if you subtract just one point from the surface of a 3-d sphere - one of its poles, say - you can in fact cover it with a suitably deformed open 2-d sphere. So the "locality" for which the surface of our sphere can be mapped to R^2 is actually pretty maximally "large". If I were Badiou, I'd be making a big deal about that subtracted point right now...).

Does this sound OK so far?
 
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Mr. Tea

Shub-Niggurath, Please
Staff member
Yep, sounds good to me. Though I must confess your maths vocab is outpacing mine somewhat with open subsets and homeomorphisms.

Also, in your last paragraph, don't you just mean a disc, rather than a 2-sphere? The surface of the earth is itself a 2-sphere, i.e. a spherical surface with dimension 2. If you include the interior of the earth, I think in geometric terms it's then considered a 'ball'. So we can visualise a 2-sphere embedded in 3-dimensional Euclidean space - and, if we wanted to, we could conceptually wrap that sphere with a (deformed) disc, which would be a filled 1-sphere (a 1-sphere is a circle), or in other words a '1-ball'.

It's impossible to visualise a 3-sphere because, if embedded in a Euclidean space, that space would have to be 4-dimensional.

The thing about wrapping a disc around a sphere is, I think, related to the delightfully named Hairy Ball Theorem.
 

poetix

we murder to dissect
Disc! That was the word I was trying to remember. Yes, a disc.

The topology stuff is me sneakily trying to shift the definition away from Deleuze's territory (differentiation, continuousness etc.) and onto Badiou's (sets, topologies, orders etc.).
 

nomadthethird

more issues than Time mag
This chapter seems good, although it's written from a critical-theoretical rather than a mathematical perspective.

This is ultimately the point of D&G's "mathematics", for what they're worth:

 

poetix

we murder to dissect
OK, on to some more detail.

So far I've described (I think) the "atlas" of a "topological manifold". The basic idea is that for each open set U of some set of open sets covering a topological space, there's a "chart", which is a homeomorphism U -> V where V is an open subset of R^n; thus the space is "locally" Euclidean even if it's not globally Euclidean. The "charts" are like my map of Northampton, projections of regions of the globe into regions of R^2; the "atlas" is a collection of charts of regions covering the entire globe. (Obviously the globe/streetmap metaphor will only get us so far here).

Wikipedia says that the underlying topological space has to be a "second-countable Hausdorff space", but doesn't say why. For the time being I'm just going to gloss these two conditions as "non-wacky".

The other thing that a topological manifold has is "transition maps", which describe how, as we move from one open set of the manifold to another, we transition between the corresponding open subsets of R^n. Is the point about a Riemannian manifold that this transition should be "smooth" as the metric measuring the distance between points on the manifold is "smooth"? For example, moving smoothly along a geodesic between two points on the surface of a sphere, I should be able to transition smoothly between the charts of the neighbourhoods of these points, like a 2-d map scrolling from left to right on the display of a satnav...
 

nomadthethird

more issues than Time mag
What's going on in parts when Deleuze uses Riemannian Manifolds (and by extension, general relativity and chaos theory), is that he is rather perversely organicizing mathematics and mathematicizing the organic by mapping terms from each over the other.

Maybe he's trying to make a Latourian hybrid out of these kinds of spaces.

(so there's smooth and striated muscle tissue, there's smooth and striated space, and they metaphorically signify the same concepts across fields for D&G without one having more authority or being more real than the other)
 
maybe a folded surface, or something like this?










not sure how to describe this object. Deleuze has influenced some amazing architecture (greg lynn, marcos novak). But the best theoretical interpretation of non-Euclidean space as a model for postmodernity has to be Massumi's in Parables of the Virtual (the chapter "Strange Horizon: Buildings, Biograms and the Body Topologic"). He equates manifolds/topologies/smooth space with affective/embodied space, which is why it can't be mapped along visual coordinates/grids. To me this aesthetic starts with the Merzbau (Schwitters), though, so it isn't postmodern at all. In fact you could argue that prehistoric art is topological/Riemannian since it tries to represent an interior or spiritual dimension.



 

poetix

we murder to dissect
Bear in mind that a topology on a set S can be just the pair of open sets S and {}. Topologies can be relatively "coarse" or "fine" - (S, {}) is the coarsest topology on S, P(S) (the powerset of S) is the finest.

A Euclidian space R^n is a set of points (R1, R2...Rn). The standard topology on such a space is that in which the open sets are the open balls around each point (plus their unions and finite intersections). Given a metric function g which measures the distance between two points p and q, an open ball of "radius" t around a point p is the set of all points for which g(p, q) < t. It's like a ball with its surface removed, so that it's all "inside". On the real line, R^1, the "open balls" are just the open intervals (p - t) < p < (p + t).
 

IdleRich

IdleRich
"Smooth" means not "striated" as in, non-Euclidean, as in, not grid-like.
I would have thought that in this context smooth means differentiable (or actually, it probably means differentiable as many times as you like).
 

Mr. Tea

Shub-Niggurath, Please
Staff member
"Smooth" means not "striated" as in, non-Euclidean, as in, not grid-like.
No, 'smooth' in this context just means continuous and differentiable, i.e. not having any sharp discontinuities.

So a sphere is smooth, whereas the surface of a cube is manifestly not smooth as it has sharp edges and vertices. A Euclidean manifold is definitely smooth, as its derivative is identically zero at all points.

Edit:

I would have thought that in this context smooth means differentiable (or actually, it probably means differentiable as many times as you like).
^ Jungian synchronicity.
 
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poetix

we murder to dissect
As often turns out to be the case, it seems that you can build up to the definition of a Riemannian manifold along a sort of tree of restrictions and extensions from a more general class of objects - viz:

Set + topology -> topological space
Topological space + Hausdorff separability and countable base -> "Non-wacky" topological space
"Non-wacky" topological space + locally Euclidean (every point has a neighbourhood homeomorphic to an open set in R^n) -> Topological manifold
Topological manifold + differentiability -> Smooth manifold
Smooth manifold + metric -> Riemannian manifold
 

nomadthethird

more issues than Time mag
No, 'smooth' in this context just means continuous and differentiable, i.e. not having any sharp discontinuities.

So a sphere is smooth, whereas the surface of a cube is manifestly not smooth as it has sharp edges and vertices. A Euclidean manifold is definitely smooth, as its derivative is identically zero at all points.

Edit:



^ Jungian synchronicity.
You're missing the point of Deleuze if you really think he cares about Riemannian manifolds. But point missing seems to be the strong suit of some people here.

I mean, I have no idea what smooth means in Riemannian manifolds, but I do have a decent idea what it means in D&G.
 

nomadthethird

more issues than Time mag
I would have thought that in this context smooth means differentiable (or actually, it probably means differentiable as many times as you like).
I dont know about the manifolds, but "smooth" is somewhat categorically used by Deleuze in opposition to "striated" and this opposition seems to loosely fit here.

"Differentiable as many times as you like" does seem consistent with "smooth" spaces a la D&G...

So you and Tea know math-- explain why RMs are relevant to general relativity?
 
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