Non-embedded Manifolds: How do they happen

[marschmellow]

I've always assumed that for a non-Euclidean manifold to exist, it has to be ambient in some higher-dimensional Euclidean space, like how a 2-sphere is ambient in 3-dimensional Euclidean space. But I've been hearing hints that higher-dimensional embedding is in fact unnecessary to define a twisty, curvy manifold, and I want to know the logical sequence behind its creation, so I know what to start rigorously learning the details of first. Since everything (not a lot) I have learned about differential geometry is self-taught, I have a farrago of concepts jumbled around in my head, and I'm not sure how they all relate.

I can totally understand how distance is naturally defined differently on a 2-dimensional sphere as opposed to ℝ^2. If you want to move from the point (x,y,z) to (x+dx,y+dy,z) in the plane, you have to move √(dx^2+dy^2) units, but if you want to make that same move in the x-y plane but on the surface of a sphere, you have no choice but to change your z coordinate too if you want to stay on the sphere, which adds a little bit of extra distance under the radical.

What I just summarized there is something that makes complete sense to me. I started with a fundamental concept of distance (the Euclidean metric) and then decided to twist the 2-dimensional subspace {(x,y,z)|z=0} of ℝ^3 into the 2-sphere. My fundamental concept of distance hasn't changed, but the actual distances that I will calculate naturally change with the way that I changed the surface.

That's what I was hoping differential geometry would be: you define some manifold (a "smooth" subset of a higher-dimensional Euclidean space) and then just see what distances naturally turn out to be using basic calculus. But apparently I'm wrong in some way.

Here is a list of concepts that I think I understand, which will be followed by my current and hopefully false understanding of what begets what:
-Metric tensor
-Topology (not the discipline, but the collection τ of subsets of a set X)
-Topological space
-Metric space
-Manifold
-"Shape" of a manifold (something I made up, probably)

Does topology induce a metric, or vice versa, or are they independent things? Does the topology even do anything or does it just sit there smiling because it satisfies 4 rules? I feel like the intuitive notion of the "shape" of a space should naturally give rise to a distance function which is fundamentally based on Euclidean distances, but the metric coefficients are in general not equal to the Kronecker Delta to account for the curving and stretching--the "shape"--of the space. Does the "shape" of the space depend on the topology, or just the set of points X?

So to summarize, I would have thought that "shape" of the manifold is determined by exactly what smooth subset of a higher-dimensional Euclidean space you're sampling from, and that shape then determines the distance function at every point which only alters the Euclidean (or Lorentzian) distance for the purpose of correcting for curvature. But it seems like what happens instead is that you just define a distance function at every point, which then induces the "shape" of the manifold, which allows you to define a manifold without embedding it. Maybe it's mathematically equivalent, but it seems backwards to me. Furthermore, what's so evil about embedding?

This is a bit of a ramble, so I want to end with a more succinct question if everything I've written is so wrong you don't know where to begin in correcting me: what should I learn first? I know that I don't rigorously understand most of what I'm talking about, but I have to start somewhere, and I want to start at the thing from which everything else branches off. Is that topology, or differential geometry, or what?

 

[mathwonk]

to answer the question in your title, they can arise as quotients of other manifolds by group actions. e.g. projective space is the quotient of R^n - {0} by the scaling action of the group R-{0}.

Also the set of all lines in R^3 is a homogeneous object, hence a manifold. I.e. start from the set of all ordered pairs of distinct points in R^3, and define an equivalence relation on them where one pair is equivalent to another if all four points are collinear.

This is the manifold of lines in R^3. The projective plane is the submanifold of lines in R^3 passing through the origin. None of these lives naturally in Euclidean space, but all can be visualized there.

 

[Jamma]

Does topology induce a metric, or vice versa, or are they independent things?

Just to answer a small part of your post- no it certainly doesn't (the reverse is obviously true though- a metric induces a topology). The metric gives much richer information.

If you are just interested about the topology of your manifold, you probably want to look at results such as this:

http://en.wikipedia.org/wiki/Whitney_embedding_theorem

 

[quasar987]

I've always assumed that for a non-Euclidean manifold to exist, it has to be ambient in some higher-dimensional Euclidean space, like how a 2-sphere is ambient in 3-dimensional Euclidean space. But I've been hearing hints that higher-dimensional embedding is in fact unnecessary to define a twisty, curvy manifold, and I want to know the logical sequence behind its creation, so I know what to start rigorously learning the details of first.

It basically comes down to point-set topology (the discipline, not the collection τ of subsets of a set X), the moral of which is 'you can talk about such things as continuity without being in a subspace of R^N!' Since an embedded n-manifold in R^N (such as curves and surfaces in R³) are defined basically as subspaces of R^N that are locally homeomorphic to R^n, then just go ahead and say that by definition, an abstract manifold is a topological space that is locally homeomorphic to R^n for some n. You might also want to add conditions to rule out bad bahavior, such as asking that the space be Hausdorff and with a countable basis (as the curves and surfaces are).

To start rigorously learning the details of abstract manifold theory, just pick up Loring Tu's 'An introduction to manifolds' or John Lee's 'Introduction to smooth manifolds' or both and start reading.

 

[lavinia]

Geometry on a manifold is determined by a metric. If the manifold is embedded in Euclidean space it inherits a metric from the embedding. But metrics can be defined without any reference to an embedding. For instance define the inner product of two tangent vectors at a point in the plane to be exp(-(x^2 + y^2)) times the usual inner product.

It is probably true that geometry started out by only looking at surfaces embedded in 3 space. So much of the mathematics dealt with the shape of the surface. But at some point ,mathematicians realized that there is a part of geometry that is intrinsic. This part can be determined by measurements on the surface only, without looking outside of it. For instance, the ratio of the circumference of a circle in polar coordinates to is radius gives you an estimate of curvature. In a positively curved world geographers would find that the radius is too large, that is 2$\pi$r is larger than the circumference of the circle.

I imagine that when non-Euclidean geometry was discovered people tried to make measurements of space to determine whether this new geometry was the actual geometry of space rather than Euclidean geometry. I think that Gauss measured large triangles to see if the sum of the angles was less that 180 degrees.

In general relativity, measurements such as these are used to detect the curvature of space-time. Near a large star, measurements of radii and circumferences show space to be positively curved. I wonder what dimensional Euclidean space is needed to contain the Universe and what shape it has inside this larger space.

 

[marschmellow]

Thank you for your responses, everybody. They collectively both cleared up some confusion and led me in the right direction for what to read next.

 

[homeomorphic]

That's what I was hoping differential geometry would be: you define some manifold (a "smooth" subset of a higher-dimensional Euclidean space) and then just see what distances naturally turn out to be using basic calculus. But apparently I'm wrong in some way.

Not completely wrong:

http://en.wikipedia.org/wiki/Nash_embedding_theorem

To some degree, it's a matter of style whether you want to be more intrinsic (not embedded) or extrinsic (do things in terms of embeddings) in your approach.

However, it's not necessary to think of surfaces as living inside some Euclidean space. After all, the Nash embedding theorem is a difficult theorem. I definitely don't have a clue how the proof goes, except that it uses functional analysis. Somehow, it must involve studying spaces of maps in R^n.

One thing that you might like to do with manifolds is cut them apart and glue them together in various ways, and it's much easier to do that abstractly without reference to an embedding.

Does the topology even do anything or does it just sit there smiling because it satisfies 4 rules?

Of course, the topology does something. The topology is just what remains when you forget what the metric is. You lose a lot of information, but not all of it. In the case of surfaces, it tells you how many doughnut holes you have. A sphere has 0, torus has 1 holes, then you can have the surface of a doughnut with two holes, and so on. The topology can help tell surfaces apart. If two surfaces equipped with metrics are isometric, they better also be homeomorphic. But maybe it's easier to tell them apart topologically than to verify directly that they are not isometric.

So to summarize, I would have thought that "shape" of the manifold is determined by exactly what smooth subset of a higher-dimensional Euclidean space you're sampling from, and that shape then determines the distance function at every point which only alters the Euclidean (or Lorentzian) distance for the purpose of correcting for curvature. But it seems like what happens instead is that you just define a distance function at every point, which then induces the "shape" of the manifold, which allows you to define a manifold without embedding it. Maybe it's mathematically equivalent, but it seems backwards to me. Furthermore, what's so evil about embedding?

It might help to think of yourself as a little ant living in the surface. The ant has no idea what the "shape" of the surface is because it is too small to know. Yet, the ant is able to figure an awful lot out just by making measurements within the surface. In a way, we are exactly in the ants position because we live on an approximately spherical Earth. It's not even obvious from the start that the Earth is even curved at all to us. It is only obvious to us because we have seen pictures from space and/or know that it is possible to travel around the world and come back to where you started. So, this indicates that shape isn't really so fundamental. You can do a lot of stuff that only depends on measurements within the surface itself that are totally independent of how that surface is sitting in space. The ant can do geometry in the surface without knowing about the shape of the surface. Geodesics are the straight lines of the geometry, the ant knows how to measure angles, and everything else goes from there.

Another way to think of it is this. If you bend a surface without stretching it, it changes the "shape", but it leaves the geometry inside the surface unchanged. Distances and angles are not affected. So, this leads you to think that the "shape" is not so fundamental. The ant in the surface can do geometry in the surface.

This is a common theme that runs throughout mathematics. Often, you have objects that initially seem to appear as being embedded in some way. They have a concrete realization. So, the approach of modern mathematics is to look at what the essential structure is that needs to be studied and then define it in a way that doesn't depend on how things are embedded. The idea is that many of these concrete realizations are actually the same or "isomorphic". So, there must be some abstract version of it that doesn't really live anywhere in particular. This is true of many mathematical objects, such as groups, rings, fields, operads, manifolds of various sorts, and the list could go on. Historically, there were many concrete examples of these things sitting around first before the abstract definition was ever found. Pedagogically, I think this should still be the case today, but often, the cart is put before the horse. Also, with many of these examples, there are embedding theorems that tell us that, actually, the abstract definition isn't really any more general because there are concrete realizations of all the abstract groups or manifolds or whatever.

This is a bit of a ramble, so I want to end with a more succinct question if everything I've written is so wrong you don't know where to begin in correcting me: what should I learn first? I know that I don't rigorously understand most of what I'm talking about, but I have to start somewhere, and I want to start at the thing from which everything else branches off. Is that topology, or differential geometry, or what?

The order I would learn it in would be first, geometry of curves and surfaces in R^3, then point-set topology, then differential topology, then, Riemannian geometry. I don't know what reference are the best here.