f122
| 22 |
|---|
The Geometric Context.
...A sphere is a two-dimensional space. Two parameters are needed to locate a point on it. It is a space that has a topology (for more details on the meaning of the word topology, see my comic book Le Topologicon, Ed. Belin). A sphere does not have the same topology, the same "shape," as a torus. A sphere has geodesics. One can inscribe a path connecting two points M1 and M2 and measure the length s traveled. This length is independent of the coordinate system chosen to locate the points, just as the geodesic curves that populate the surface are.
...Connect the center of this sphere to all its points. We obtain an infinite number of half-lines. These can be labeled using the same coordinate system as the points, for example two angles q and j.
Above, our sphere. We have made a hole to show the set of vector rays.
Now remove the sphere and keep only the vector rays.
...We have truncated these half-lines, but in fact, they are infinite. Each is defined solely by two parameters, for example two angles. The metric structure has disappeared. No more geodesics, no more lengths. What remains?
-
Every half-line has a neighborhood. One can select neighboring half-lines to enclose this one within a kind of cone. Inside this cone, one can place an even narrower cone containing the half-line. This is similar to concentric circles or Russian dolls, but with bundles of half-lines. However, we are not drawing geodesics on these cones. Each of their generators is simply a set of two parameters, for example two angles.
-
An intuitive idea of differentiability is preserved. There is no discontinuity in this "texture."
Take a flat surface, with geodesics, length, etc.
...No matter which coordinate system I choose, I will always need two real numbers (x,y), (r,q), etc., to locate my points.
These real numbers come from R², that is, from the set of pairs of real numbers, such as (3.8705, -17.56). Any pair of points taken from this space of real number pairs has a neighborhood. It is "continuous."
These "pre-metric" objects are called manifolds (mathematicians have a knack for choosing words that evoke nothing at all for the average person).
...At this stage, one can thus skip the step of considering a set of n real numbers (an n-dimensional space) without automatically attaching ideas of length or geodesics to it.
...It is somewhat like considering a surface whose points are constrained only to maintain contact with their neighbors. It would be infinitely elastic and deformable. By convention, if we represent a surface by its boundary (either its edge or its apparent contour), we evoke this "moving" concept of manifold by simply removing the boundary:
...This image, in fact, evokes the shadow of the object. And a shadow has neither substance nor form. Its geometry depends on the object onto which it is projected.
One can also imagine a manifold (in English, manifold), without its metric, as a family of lines.
...Here, the lines appear parallel. But they should actually be arranged in any way, as long as their proximity and neighborhood relationships are preserved.
...Ultimately, a good image of a 2-dimensional manifold V2 is a bundle of spaghetti that is first cooked, then bent and twisted in every direction, without altering the order of the strands relative to one another.
In any case, one can perform a two-sheeted covering operation on a manifold, equipping each sheet with metrics, as suggested by the image below:
Here, two 2D sheets equipped with identical (Euclidean) metrics. But one could equally well have:
...We will call M and M* conjugate points. The fact that the two conjugate spaces are constructed as a two-sheeted covering of a manifold simply means there is a point-to-point correspondence between the two sheets F and F*, but, for example, the distances between corresponding pairs of points (M1,M2), (M1, M2) may differ. The only constraint is that neighborhoods of points correspond to one another, and that every non-singular region of one sheet corresponds to a non-singular region of the other.
...We return to the flexible bundle of noodles from earlier. The "manifold-skeleton" structure exists solely to establish an injective mapping between the two geometric objects. The above diagram is intended to completely dissolve questions such as "How are the sheets F and F* arranged relative to each other? If F is a universe, where is F*?" These sheets are simply conjugate, with a point-to-point correspondence, and these conjugate points can be described using the same coordinates.
../../../bons_commande/bon_global.htm
Table of Contents
article Table of Contents
Science Home Page
Number of views of this page since July 1, 2004**: