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Link to thread of next article. &Interpreting Curved 3-Manifolds With Color. Date: 1997/12/16 I have on and off been trying to get a feel for curved 3-manifolds. I have thought of something that may be helpful. Consider some the ways in which the height of the earth's surface above or below some mean value can be conveyed on a map. The two primary means are by the use of contour lines or color. I think that the same technique may be used for 3-dimensional manifolds.
Just as a curved surface in R3 can be projected onto R2 I think a curved 3-manifold M in R4 can be projected onto R3. Consider a small region S of such a projection. Then there exists a function on S, f(x,y,z) that gives the value of the forth coordinate w.
Now just as the curvature of a surface can be conveyed on a 2-dimensional map using color so can color be used on a "3-dimensional map" to convey the curvature of a 3-manifold. What seems interesting is the idea that this scalar function f contains all information regarding the curvature of the manifold M. Let us do a Taylor series expansion of f about some point X0 = (x 0,y0,z0) then we may make some connection between the expansion coefficients and the 6 independent components of the Riemann tensor for a curved 3-manifold?
I suspect the last six differentials above might somehow be related to the six independent components of the Riemann tensor for a curved 3 manifold?
A pretty curved manifold -> Sin[x]Sin[y]Sin[z]Sin[t] Date: 1998/01/25
Consider the 1-dimensional curved manifold,
where x and z are the familiar coordinates of flat 2-space. Isn't she pretty, a 1-dimensional curve imbedded in flat 2-space.
Consider the 2-dimensional curved manifold,
where x, y, and z are the familiar coordinates of flat 3-space. Isn't she also pretty, a 2-dimensional curved surface imbedded in flat 3-space.
Make a topographic map of this function. This can be done with Mathematica with the following "code",
With such a plot and a little thought one can pick out regions of positive, negative and zero Gaussian curvature.
Now lets bust out of flat 3-space and consider the 3-dimensional curved manifold,
where w, x, y, and z are the familiar coordinates of flat 4-space. Isn't she a beauty, a 3-dimensional curved manifold imbedded in flat 4-space.
Now consider a 3-dimensional "topographic" map of this manifold. Now instead of one-dimensional contours one has two-dimensional contours. (what follow are several slices of w = Sin[x]Sin[y]Sin[z])
For fun let w above depend on time, say,
Now for fixed time the distance squared between two near points X and Xo is,
Using this and a three-dimensional Taylor series expansion of w one can determine the metric g_ij for this manifold (we are working on this step so don't take my word for this). With the metric g_ij in hand get a copy of "Mathematica For Physics" by Robert L. Zimmerman and Fredrick I. Olness. Chapter 7 of this book has the simple Mathematica code needed to determine all the important tensors of General Relativity given the metric g_ij. In particular one might want to determine the stress-energy tensor of the manifold represented by w above even though it is probably of no relevance to our Universe.
So take that simple scalar or complex scaler function you have and consider the pretty curved manifold it represents ( for example psi = exp[i(wt-kx)] can be thought of as a pair of curved 3-manifolds which change with time ).
S^1, S^3, and S^7 are parallelizable. Was Re: QFT. Date: 1998/01/29
In article <6a8o8d$16u8@r02n01.cac.psu.edu> ale2@psu.edu (ale2) writes:
> In article <6a830j$25p@gap.cco.caltech.edu> > kevin@cco.caltech.edu (Kevin A. Scaldeferri) writes: > > > This is related to > > various neat facts like that the only spheres which are parallelizable > > are S^1,S^3 and S^7. (Of course you have to know what parallelizable > > means and why it is an interesting property of a manifold to > > appreciate this fact.) > > This sounds like an interesting fact. If its not too much trouble could > you explain parallelizable.
I asked this question and i did not realize the answer was on my shelf, shame on me.
"Geometry, Topology, and Physics" by Mikio Nakahara
"Topology and Geometry for Physicists" by Charles Nash and Siddhartha Sen.
From page 220 of Nakahara we learn,
"If an m-dimensional manifold M admits m vector fields which are linearly independent everywhere , M is said to be parallelisable"
That S^1 is parallelisable is easy to see, that S^2 is not is a little harder and gives rise to that theorem about not being able to comb all the hair on a ball (how does that go?). Now that S^3 is parallelisable might strain ones imagination so here are three orthonormal vector fields on S^3 which satisfy the above definition of parallelisable. (The following is also from page 220, Nakahara.)
Nakahara goes on to show "how this parallelisability of S^3 is related to the existence of quaternions", for you quaternion fans, but you probably know this.
Anyone know some interesting fun facts (theorems) about manifolds which are parallelisable? In particular S^3.
For example, take one of the vector fields above, say e_1. Now pick some starting point and go always in the direction e_1. Is it easy to show that one will or will not come back to ones starting point?
Mapping manifolds with torsion to flat manifolds with no torsion. Date: 1998/01/30
Consider a three-dimensional, closed, isotropic, homogeneous manifold with torsion. Can such a manifold be mapped in a simple way to a "flat" manifold with no torsion in say the way S^3 can be mapped to R^4? Any examples of such mappings?
Min of some measure of N points in S^3 and S^4. Date: 1998/03/02
The recent post about minimizing some measure of N points on a sphere jolts me into wanting to know if anything interesting happens when the space in question is S^3 or S^4. Any basic papers or articles on this?
If point --> line; S^2 --> S^3 and we try to find some minimum measure of N "straight" lines in S^3 do we get an interesting problem?
Re: min of some measure of N points in S^3 and S^4. Author: Dave Rusin <rusin@vesuvius.math.niu.edu> Date: 1998/03/03Forum: sci.math
In article <6degm6$1ad4@r02n01.cac.psu.edu>, ale2 <ale2@NOSPAMpsu.edu> wrote: >The recent post about minimizing some measure of N points on a sphere >jolts me into wanting to know if anything interesting happens when the >space in question is S^3 or S^4. Any basic papers or articles on this?
I don't know if this has been pursued.
Oh, sure, you can try to arrange N point on other sets too, but by and large there isn't a fixed way to do this because of the same problem: there's no single best way to describe what it means to be spaced uniformly. As long as you have a metric space you can try something like maximizing the minimum distance between two of the N points, or the sum of the squares of the reciprocals of the interpoint distances. But is there any interesting space other than the round circle on which the various notions could be expected to coincide for all N? I don't know of any.
Once the minimizing criterion is selected, one could compute (locally) optimal configurations for any N on any Riemannian manifold by following flows. I don't know if there's anything exciting to expect.
>If point --> line; S^2 --> S^3 and we try to find some minimum measure >of N "straight" lines in S^3 do we get an interesting problem?
Um, there aren't any lines in S^3; do you mean circles? You could look for arrangements of great circles in S^n by observing that each is the intersection with S^n of a 2-dimensional plane in R^(n+1); thus the family of things you're shifting around is the set of points in the Grassmannian manifold of 2-planes in R^(n+1) (a fairly sphere-like topological space!).
As above, one may ask about finding an optimal arrangement of points in the Grassmannian (=great circles on the sphere) only after a choice of optimality criteria has been made; but of course that has to follow the choice of a metric on this manifold. I don't know if there's an obvious choice of metric (in either the metric-space sense or in the sense of Riemannian geometry). I've always worked in the topological category.
There's nothing to stop you from considering the other Grassmannians, of course; if you can resolve the questions of metric and objective function, you can ask about "well-spaced" collections of N great k-spheres on S^n (= collections of N (k+1)-dimensional subspaces of R^(n+1) ).
dave
Re: min of some measure of N points in S^3 and S^4. Author: Les Reid <reidl@mast.queensu.ca> Date: 1998/03/06Forum: sci.math
Regarding arrangements of points on Grassmannians, see "Packing Lines, Planes, etc.: Packings in Grassmannian Spaces" by J.H. Conway, R.H. Hardin, and N.J.A. Sloane in Experimental Mathematics, v.5 (1996), #2, pp 139-159. The bibliography also contains references to the problem of packing points in S^n.
Does S^3 admit a "cubic" lattice of points? Date: 1998/03/04
Does the fact that S^3 is parallelizable mean we can lay out a "cubic" lattice of points in S^3? Cubic in the following sense, if we start at one point of such a lattice there will be three orthogonal directions, say i, j, and k, such that if we go a distance C/N in either of these directions we will come to another lattice point (C is the circumference of S^3 and N is some integer).
If so it seems like we could have all kinds of fun considering defects of such lattices?
Re: Does S^3 admit a "cubic" lattice of points? Author: Charles H. Giffen <chg4k@virginia.edu> Date: 1998/03/05Forum: sci.math
ale2 wrote:
> Does the fact that S^3 is parallelizable mean we can lay out a "cubic" > lattice of points in S^3? Cubic in the following sense, if we start at > one point of such a lattice there will be three orthogonal directions, > say i, j, and k, such that if we go a distance C/N in either of these > directions we will come to another lattice point (C is the > circumference of S^3 and N is some integer). > > If so it seems like we could have all kinds of fun considering defects > of such lattices? > > Thanks for any ideas!
View S^3 as the unit quaternions, and consider the "lattice" as the collection of the eight points: 1,i,j,k,-1,-i,-j,-k. If you take any such point, x, and multiply it by i, j, or k, obtaining ix, jx, or kx, respectively, then you land back in the "lattice". The geodesic distance one "travels" (in S^3 viewed as the unit sphere) for each such step is pi/2 -- precisely one-quarter of a great circle.
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