From: rusin@vesuvius.math.niu.edu (Dave Rusin) Newsgroups: sci.math Subject: Re: Question about non-euclidian geometry Date: 19 Aug 1996 22:01:50 GMT In article <4usqsc$mnn@news.indy.net>, Greg Moriarty wrote: >Usually to grasp the concept of non-euclidian geometry the student is told >to imagine a 2 dimensional plane wrapped around into a 3 dimensional >sphere. To the ant living on the plane, to which the 3rd dimension is a >foreign concept, parallel lines seem to intersect and triangles add up to >>180 degrees and other such silliness. To the god-like observer, those >things aren't odd at all since the god-like creature has an understanding >of the 3rd dimesion. My question is thus: > >Can one go from a non-euclidian geometry to a euclidian geometry by >introducing a new dimension? Is this something that is done? If so, can >one ALWAYS do so? and later... >On a side note, I remember a class called differential geometry >in college. Does this deal with non-euclidean goemetry? Is >N-E geometry a part of differential geometry? Are they synonymous? You have to tell us just what "geometry" means. Differential geometry is the study of Riemannian manifolds, roughly, sets which may locally be deformed into a Euclidean space, and which allow the measurement of lengths, but not perhaps in a way consistent with the deformation. Your ant, above, thinks the portion of the sphere close to it looks like a portion of the plane. The ant can also measure distances by walking along curves, seeking the shortest possible path joining two points. But a clever ant would note that the one-to-one correspondence between its little neighborhood and a portion of the plane does not preserve distances, angles, and lengths (hence the frustration of global map-makers). In this context, what you appear to be asking for is true: there is a theorem of Nash (recent recipient of the Nobel Memorial prize in Economics) which states that every (suitable) Riemannian manifold is isometric with (= "the same as") a subset of some higher-dimensional Euclidean space. (I think the dimension of the latter is quadratic in the dimension of the original). Note that Euclidean space R^n itself is also an Riemannian manifold under the usual metric, so no, "differential geometry" does not really presuppose "non-Euclidean geometry". On the other hand, there are other interpretations of "geometry" for which this discussion is inappropriate. For example, finite group theorists often make use of "geometries", which are simply sets of sets with certain incidence properties. In this direction I might also mention the projective geometries (and affine geometries) which more or less correspond to replacing the real numbers with other fields (or rings) but still using coordinates; now "distance" is not really defined. dave ============================================================================== From: rusin@vesuvius.math.niu.edu (Dave Rusin) Newsgroups: sci.math Subject: Re: Question about non-Euclidean geometry Date: 20 Aug 1996 18:54:47 GMT I would like to clarify a couple of responses to the original post, which began: In article <4usqsc$mnn@news.indy.net>, Greg Moriarty wrote: >Usually to grasp the concept of non-euclidian geometry the student is told >to imagine a 2 dimensional plane wrapped around into a 3 dimensional >sphere. To the ant living on the plane, to which the 3rd dimension is a >foreign concept, parallel lines seem to intersect and triangles add up to >>180 degrees and other such silliness. To the god-like observer, those >things aren't odd at all since the god-like creature has an understanding >of the 3rd dimesion. My question is thus: > >Can one go from a non-euclidian geometry to a euclidian geometry by >introducing a new dimension? Is this something that is done? If so, can >one ALWAYS do so? I responded to this (and some later points) in a post which might appear to be at variance with another follow-up, which I feel sort of misses the point. In article <4vagvb\$hpl@mountaindew.eng.umd.edu>, Kevin Anthony Scaldeferri wrote: >There is a theorem which says that any manifold satisfying the conditions >(mumble,mumble) can be embedded in a Euclidean space of sufficiently >large dimension. (Specifically 2N+1, where N is the dimension of the >manifold in question). As I recall the conditions are not very >stringent, possibly just that the manifold be differentiable. Any >topology text should have the exact statement. (Whittney Embedding >Theorem). I suggest Guillemin and Pollack as a fairly accessible book >to find out more, if you are so inclined. This gives an opportunity to explain a situation which arises quite frequently in geometry and topology (and elsewhere in mathematics), namely, what does it mean to call two things "the same"? Usually this means that one can set up a one-to-one correspondence between the points in the two objects, such that this correspondence (and its inverse) "preserve the structure" of the things. But then one must ask just what structure is to be preserved. (Keywords: Category Theory). Let's begin with Moriarity's ant. The ant is convinced that it lives on a 2-dimensional plane, and indeed demonstrates this by establishing "rectangular" coordinates on the sphere -- usually, latitude and longitude. All is well unless the ant is well-traveled: when crossing the meridian opposite the Greenwich meridian, one coordinate suddenly switches from +180 to -180 or vice versa. If the ant believed that it lived on the plane, it would have to conclude it had suddenly jumped from one edge of a rectangle to an opposite edge. Clearly, then, our coordinates have not demonstrated that the ant lives on the plane, but they are still useful. Indeed, even the ants near this "intercolonial date-line" would have no trouble establishing coordinates. They might simply agree to measure longitude from 0 to 360 -- from their perspective, it's the ants near Greenwich which have trouble. Tourists from the two regions simply understand that when traveling to the other part of the sphere, one changes coordinates by adding or subtracting 360 degrees (in some cases) to the longitude. There is a more serious problem near the poles, where latitude and longitude reduce, more or less, to polar coordinates near the origin in the plane. In particular, there is no longitude which can be assigned to the poles in a way which would vary continuously near the poles. Of course, residents near the poles can simply lay out a rectangular grid of coordinates near the poles; they would have no problem locating a point by giving it's coordinates "up from Main Street and over from First Avenue". Again, tourists would have to be willing to convert coordinates when traveling. This time the change-of-coordinates calculations are more complex, requiring the familiar trigonometric functions, but at least at all points where both sets of coordinates are defined, the transformation rules are differentiable functions. With this set of four coordinate systems, the ants have collectively proven that they live on what we call a _manifold_: a (Hausdorff) topological space with the property that every point has a neighborhood homeomorphic to Euclidean space. Moreover, their choice of coordinate systems has the property that the change-of-coordinates functions may be differentiated k times (with continuous k-th derivatives) for any k = 0, 1, 2, ..., (even k=infinity, that is, the functions are real-analytic). Thus we say that the manifold they have set up is a _differentiable manifold_ or a _C^k-manifold_. One easy way to get a manifold is to take a subset M of Euclidean space R^n which has the property that near each point p of M one may choose a new coordinate system for R^n such that the points of M which are near p are precisely the points of R^n near p for which a few of the coordinates are zero. We know, for example, that the unit sphere S inside R^3 may be described in this way, if for all points on R^3 excluding the origin we use spherical coordinates. (I suppose I have to use r-1, theta, and phi, actually, to be consistent with the opening sentence: then S is exactly the set of points where the first coordinate is zero.) Manifolds created in this way are called _submanifolds of R^n_. Again one can ask about the smoothness of the transition functions between the new coordinates selected for neighborhoods of R^n, relative to the standard coordinates; these define the smoothness of the submanifold. Now to the Whitney theorem(s). Suppose the ants in various parts of a planet managed to set up coordinate systems near their homes. Each hangs a map up in the living room, showing "home" in the center, a big square around it, and at the edges some darkness with the message "Here be anteaters". A planetary conference is called, and amazingly, each ant's map is found to overlap with it's neighbors' just a little, and the coordinates each one uses can be converted to it's neighbors' coordinates just by adding or subtracting a constant to one or both coordinates (just as the Greenwich and anti-Greenwich crowds do). But something unusual emerges: When the Greenwich and anti-Greenwich crowds on earth superimpose their maps along the overlaps, they find a portion of the earth repeated on the left and right edges -- once each from each mapmaker's map. On the other hand, these ants notice that when all the maps are superimposed, there is some repetition along all four edges. But what shows up at the bottom left is repeated at the top _right_; the rest of the left edge from bottom to top is repeated on the right from top to bottom. Likewise, the top edge repeats the bottom edge but with the orientation reversed. The fact that Euclidean coordinates are established at each point shows that the planet is a manifold (indeed, since the change-of-coordinate maps are translations, it's a C^k manifold for all k). But it looks different from the plane, and it doesn't appear to be the sphere either. Indeed it's not: one can show that there is no continuous one-to-one correspondence between the points on their planet and the points on the sphere. So what does the planet "look like"? A good picture might be found if we could find a submanifold of Euclidean space to which this planet is homeomorphic ("the same"). Some experimentation might lead you to despair, since, as it turns out, there is _no_ submanifold of R^3 which is the same as this planet. In 1935, Hassler Whitney showed that every C^infinity manifold M of dimension n may be embedded in R^N as long as N > 2n, that is, there is a submanifold of R^N whose points may be put into 1-to-1 correspondence with M in a continuous way (with the inverse also continuous). Actually, the proof can be made simpler if you don't care how big N is, but there's some fascination with the minimal embedding dimension. Indeed, Whitney expended considerable effort improving the theorem to say " ... N >= 2n " in 1944. In particular, our ants, living on an n=2 - dimensional planet can view themselves as living on a subset of R^4. (Their manifold is known as "Projective 2-space" or "the projective plane"). I'm sorry, I don't know offhand a concise description of such a subset of R^4 in this way, but here's an embedding of their planet into R^5: it's the subset defined by the equations (1-x1) x1 = x3^2 + x5^2 (1-x2) x2 = x4^2 + x5^2 x1 x2 = x5^2 (Those in the know will recognize Plucker coordinates; use the map f: (a, b, c) -> (b^2, c^2, ab, ac, bc) to get a two-to-one map from the 2-sphere into R^5.) . . . . . . . . . . . . . . . . . . . . But is this really "the same" as their planet? Indeed, is the sphere really "the same" as earth? If all you want is a one-to-one correspondence which is continuous, then yes, the earth is a sphere. But we know the earth bulges at the equator; it has mountains and valleys; certainly, it is not of radius 1 if we use any conventional units. This difficulty is noticed even by earthly ants: although they can easily set up coordinates around their homes and make maps, these maps fail to preserve angles and lengths. If we use latitude and longitude to equate portions of the earth with portions of the Euclidean plane, then paths which on the map appear to be the shortest among all those joining two points are often _not_ the shortest as measured by walking on the earth. Now the ants are considering _Riemannian manifolds_. These are manifolds in the above sense but with the additional structure that at each point in the manifold we have not only a one-to-one correspondence with Euclidean space, but we have decided to use that one-to-one correspondence to determine the inner products (dot products) of vectors at the point. In particular, we now have a notion of _lengths_ of vectors and _angles_ between vectors. (Since this is all relative to a point, computing the length of a _curve_ requires integrating direction vectors along a curve and so on, which makes the subject rather heavy reading.) Of course, Euclidean space is a Riemannian manifold, and so on any submanifold one can use the inner product of the surrounding space as the inner product on the submanifold. So, sure, you can obviously set up a smooth one-to-one correspondence between the points of a sphere and the points of an ellipsoid (both of them submanifolds of R^3), but you can't do so and hope to preserve angles and lengths; they are _the same_ as differentiable manifolds but _different_ as Riemannian manifolds. A mapmaker can convey all the information available in a Riemannian manifold. For example, a Mercator projection of (a portion of) the Earth gives a continuous one-to-one correspondence of points, and even manages to preserve angles, but it must be equipped with a little graph which shows a factor by which lengths must be reduced according to latitude. So perhaps Moriarity's question was intended to include this metric information. Suppose we ask whether every Riemannian manifold M is _isometric_ to a submanifold of Euclidean space. This asks if there is a submanifold M' of R^n such that a smooth 1-to-1 correspondence between M and M' may be set up, which also preserves lengths and angles. Using Whitney's theorems, one can certainly find an M' in R^n and the smooth correspondence, but there's no guarantee that the metric information will be at all correct. In 1954, John Nash proved that if M is a (compact) n-dimensional Riemannian manifold, and has any embedding at all in R^N, then it also has an isometric embedding in R^N (assuming N > n). Combined with the Whitney theorems, this shows M is isometric to a submanifold of R^(2n). (He also proved that non-compact manifolds can be isometrically embedded, but I don't remember what this does to the embedding dimension.) There is a technical problem here related to smoothness. Nash showed that there is indeed an isometry f : M --> M' which is one-to-one and even differentiable, but not necessarily as smooth as M. He improved this in 1956 to show that if M is endowed with a C^k metric, then there is a C^k isometry f: M --> M' to a submanifold M' of R^N as long as N is at least n(3n+11)/2. This result applies only for k > 2, but it does apply to k=infinity, that is, real-analytic manifolds may be real-analytically embedded. Non-compact manifolds may be handled as well; we need N >= n(n+1)(3n+11)/2 in that case. There has been considerable work since then as well, although I would say the problem of the minimal embedding dimension is rather more difficult than in the non-metric setting. (Here perhaps the experts can help me: it would appear from this that there is an isometric embedding of the _flat_ torus M = S^1 x S^1 into R^3, using a map which cannot be expected to be twice-differentiable. Can such an embedding be given explicitly? I don't need the Nash theorem to find a smooth isometric embedding of M into R^17, thank you.) All the papers above appeared in the Annals of Math, so as you may surmise they were and are considered deep and important. Both Whitney and Nash are fascinating characters. A great deal of attention was focused on Nash about a year ago (e.g. articles in the Intelligencer) when he was awarded the Nobel Memorial prize in Economics for some game theory work he did in the 1950s. (His proofs are deeply rooted in functional analysis, not the geometry or game theory one might expect.) The study of manifolds is Differential Topology; the study of Riemannian manifolds is Differential Geometry. Each has become incredibly robust, with many directions of research, fascinating examples, and significant applications. The material is not easy, and requires a good background in topology and analysis. Spivak's "Comprehensive Introduction to Differential Geometry", a mere 3000 pages, is an excellent resource starting at the undergraduate level. dave ============================================================================== From: ksbrown@seanet.com (Kevin Brown) Newsgroups: sci.math Subject: Re: Question about non-euclidian geometry Date: Wed, 21 Aug 1996 06:47:37 GMT keylime@indy2.uucp (Greg Moriarty) wrote: > Can one go from a non-euclidian geometry to a euclidian geometry by > introducing a new dimension? Is this something that is done? If so, > can one ALWAYS do so? Since you specified introducing just _a_ new dimension (singular), the answer is no. This was proved around 1901 by Hilbert, who showed that the original non-Euclidean space (the 2D hyperbolic plane of Lobachevski, Bolyai, et al) cannot be embedded in its entirety in 3D Euclidean space. However, it CAN be embedded in 4D Euclidean space. As others have mentioned, you can always embed a smooth metrical non-Euclidean space in a higher-dimensional Euclidean space, but it usually takes more than just one extra dimension. > I would think such a question would have practial implications. For > example, Minkowski introduced a 4th dimension to give SR a geometric > description. Einstein went a step further by making that 4-D space > non-euclidean. Could one introduce a 5th dimension to return to a > Euclidean universe? The reference to Minkowski complicates the question a bit. If you're willing to allow imaginary "distances" (i.e., distances whose squares are negative), then it IS possible to embed the hyperbolic plane in 3D Euclidean space - as the surface of a sphere with imaginary radius. In fact, as early as 1766 Lambert observed that if you assumed there are at least two distinct lines through a given point that don't intersect a given line, then the area of a triangle with angles a,b,c would be -R^2 (a+b+c-pi) for some constant R. He knew that the area of a triangle on a real sphere of radius R was R^2 (a+b+c-pi), so he wrote "one could almost conclude that the new geometry would be true on a sphere of imaginary radius". It turns out that if you just plug in iR as the radius in any formula for spherical geometry you get the corresponding formula for hyperbolic geometry. On the other hand, it isn't clear that a formally Euclidean space with imaginary distances is any more intuitive than a curved space with strictly real distances. In fact, I would argue that the most un-intuitive aspect of the relativistic "metric" of spacetime is not it's curvature, but the fact that it isn't really a metric at all. A metric space, in the strict sense of the term, is a manifold that satisfies the triangle inequality, which is the property that leads to our intuitive impressions of "locality". In particular, locality is transitive in a metric space, meaning that if A is close to B, and B is close to C, then A can't be too far from C. Spacetime doesn't satisfy this condition. There exist points A,B,C such that the absolute distances AB and BC are both less than the Planck length (10^-35 meters), and yet the distance AC is the radius of the observeable universe. This has nothing to do with curvature; it's strictly a consequence of the fact that spacetime is not a metrical space and does not satisfy the triangle inequality. In view of this, a little bit of curvature should be the least of your worries. ____________________________________________________________ | /*\ | | MathPages / \ http://www.seanet.com/~ksbrown/ | |____________/_____\_________________________________________| ============================================================================== ****************************************************************************** **************** UNPOSTED ************************************************** ****************************************************************************** I'd like to clarify a few issues of my post above. First let me acknowledge James Dolan's objection to my one-line summary of differential geometry. Surely historical precedent dictates that the right response be, "Differential geometry is what differential geometers do"; equivalently it's just AMS Subject Classification 53. My Columbia Encyclopedia (with Hyman Bass overseeing Mathematics entries) gives it as the "branch of geometry in which the concepts of the calculus are applied to curves, surfaces, and other geometric entities". Second, I'd like to describe better the planet on which the second set of ants live -- the projective plane RP^2. As I pointed out, this surface cannot be embedded in R^3, but it can be _immersed_ in R^3. (Immersions are maps which are locally homeomorphisms but which allow for self-intersections -- the standard picture of a Klein bottle is an immersion.) The classic immersion is the so-called Boy's surface. You can find some pretty pictures and parametric equations at the Geometry Center, URL: http://www.geom.umn.edu/zoo/toptype/pplane/ The map (f,g,h) describing the Boy's surface there is one-to-one except along a curve of double points; thus the map (f,g,h,k) : S^2 --> R^4 gives an embedding of the projective plane into R^4 as long as k is a homogeneous polynomial of even degree which separates collapsed points along that curve. Third, please ignore all references to real-analyticity in my previous post. Nash did considerable work in the analytic and algebraic categories, but I'm not qualified to say what it was. I must have been very tired when I confused C^\infinity ("smooth") with C^\omega ("analytic"). Finally, I still seek a description of a flat submanifold of R^3 which is diffeomorphic to the flat torus. ============================================================================== EQUATIONS FOR BOY'S SURFACE Let F = ( f(x, y, z), g(x, y, z), h(x, y, z) ) be a map from R^3 to R^3. If the functions f, g, h are homogeneous polynomials of even degree >= 2, then for all (x, y, z) in R^3 we have F(x, y, z) = F(-(x, y, z)). Thus the restiction of F to the unit 2-sphere sends antipodal points to the same image point--hence, F defines a new map from RP^2 to R^3. In the case of Boy's surface, f(x, y, z) = [ (2x^2 - y^2 - z^2)(x^2 + y^2 + x^2) + 2yz(y^2 - z^2) + zx(x^2 - z^2) + xy(y^2 - x^2) ]/2 g(x, y, z) = (Sqrt(3))/2 [ (y^2 - z^2)(x^2 + y^2 + z^2) + zx(z^2 - x^2) + xy(y^2 - x^2) ] h(x, y, z) = (x + y + z)[ (x + y + z)^3 + 4(y - x)(z - y)(x - z)] Pictures may be created parametrically in Maple by the following commands: x:=cos(t)*sin(s); y:=sin(t)*sin(s); z:=cos(s); f:=1/2*((2*x^2-y^2-z^2) + 2*y*z*(y^2-z^2) + z*x*(x^2-z^2) +x*y*(y^2-x^2)); g:= sqrt(3)/2*((y^2-z^2) + z*x*(z^2-x^2) + x*y*(y^2-x^2)); h:=(x+y+z)*((x+y+z)^3 + 4*(y-x)*(z-y)*(x-z)); plot3d( [h/8,f,g], s=0..Pi, t=0..Pi) _________________________________________________________________ Up: Boy's Surface _________________________________________________________________ [LINK] The Geometry Center Home Page [LINK] Topological Zoo Welcome Page Comments to: webmaster@geom.umn.edu Created: Jun 27 1995 ---- Last modified: Wed Aug 16 22:36:48 1995 Copyright © 1995 by The Geometry Center, all rights reserved