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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 strip Le Topologicon, Ed. Belin). A sphere does not have the same topology, the same "shape" as a torus. The sphere possesses geodesics. One can inscribe on it a path connecting two points M1 and M2 and measure the length s traveled. This length is independent of the coordinates chosen to locate the points, as are the geodesic curves that populate the surface.
...Let us join the center of this sphere to all its points. We obtain an infinity of half-lines. These can be located with the same locating system as the points, for example two angles q and j.
Above, our sphere.
A hole has been made to show the set of radius vectors.
Let us now remove the sphere and keep only the radius vectors.
...These half-lines have been truncated, but in fact they are infinite. Each is defined only by the giving of 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 it in a sort of cone. Inside this cone one can place a narrower one, which contains the half-line. This is a matter of concentric circles or Russian dolls, but with bundles of half-lines. But it is not a matter of drawing geodesics on these cones. Each of their generatrices is simply a set of two parameters, for example two angles.
- One distinguishes an intuitive idea of differentiability. There is no discontinuity in this "texture."
Let us take a plane surface, with geodesics, length, etc...
...Whatever coordinate system I choose, I will always have to locate the position of my points with two real numbers (x,y), (r,q) etc...
These real numbers are taken from R2, that is to say from the set of pairs of real numbers, such as (3.8705 , - 17.56). Every pair of points taken in this space of pairs of real numbers has a neighborhood. It is "continuous."
These "pre-metric" objects are called manifolds (mathematicians have a knack for choosing words that have no evocative power for the man in the street).
...At this stage one can therefore take this step, which consists in considering a set of n reals (n-dimensional space), without automatically attaching to it any idea of length or geodesics.
...It is a bit as if one were considering a surface whose points would have no constraint other than to keep in contact with their neighbors. It would be infinitely elastic and deformable. By convention, if we represent a surface by its contour (either its edge or its apparent contour, we will evoke this "moving" concept of manifold by simply removing the contour:
...This image moreover evokes the shadow of the object. And a shadow has neither consistency nor shape. Its geometry depends on the object onto which it is projected.
One can also represent the manifold (in English manifold), without its metric, as a family of lines.
...Here, lines have been put that look parallel. These lines would need to be... anyhow, while preserving their relations of proximity, of neighborhood.
...Finally, a good image of a manifold V2 is a bundle of spaghetti that one first cooks, then can fold and twist in all directions, but without modifying the arrangement of the strands relative to one another.
Be that as it may, one can perform on a manifold a two-sheeted covering operation, which one equips with metrics, suggested by the image below:
Here two 2d sheets endowed with identical (Euclidean) metrics. But one can just as well do:
...M and M* will be called conjugate points. The fact of saying that the two conjugate spaces are constructed as the two-sheeted covering of a manifold simply means that there exists a point-to-point correspondence between the two sheets F and F*, but, for example, the distances between pairs of homologous points (M1,M2) , (M*1, M*2) can be different. The only constraint is ultimately that the neighborhoods of points also correspond and that to every non-singular region of one sheet corresponds an equally non-singular region.
...We find again the bundle of flexible noodles from earlier. The structure of "skeleton-manifold" is there only to construct the injective mapping between the two geometric objects. The drawing above is intended to make questions like "how are the sheets F and F* arranged relative to each other? If F is a universe, where is F* ?" completely explode. These sheets are simply conjugate, with a point-by-point correspondence, and these conjugate points can be described by the same coordinates.
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