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Geometric context.****
A sphere is a two dimensional geometric object. We need two quantities, two numbers, two scalars, to locate a point on it.
A sphere is a surface which owns a topology. Its topology is different from the torus' one.
Both own geodesic systems. As pointed out in a preceding section, we can figure two distinct points M1 and M2 on a sphere and a curve which links them. Then we can measure the length along this peculiar path. It's a coordinate-invariant quantity. A S²-sphere exists independently of any 3D representative space. But we can figure it in our familiar Euclidean 3D space, which we are supposed to live in. Then we can give a center to this sphere and join all its points to it. See figure (116). Each point corresponds to two angles: q and j.
(116)
We have made a hole in the sphere in order to show the vectors OM, where O is the center and M a point of the sphere.
Now, figure (117) keeps the vectors and forgets the sphere.
(117)
These half straight lines are infinite, but we have represented them cut at a given length, corresponding to the radius R of our sphere. Each straight line corresponds to a couple (q, j). The metric structure has disappeared. No geodesics, no length. What's remaining?
Any of those half straight lines has neighbors, which form its neighborhood. Each half straight line could be imagined as enclosed in a succession of cones (figure (118)).
(118)
Around any line we can put as many cones as we want. Between two of these cones we always can put another one. It suggests intuitively the concept of differentiability. In such a geometrical object there is no discontinuity.
Now, forget the sphere and take a plain surface. It is a set of points. Whatever is the coordinate system I choose, I can define the points with two quantities: (x,y), (r,q), and so on.
A couple of real numbers. These are picked in R², i.e. in the set of real numbers, like (3,8705, -17,56).
Any couple of real number (x; y) has an infinite number of neighbors (x + Dx; y + Dy).
These "pre-metric" objects are called by mathematicians manifolds.
It is rather difficult to think about such "flexible" medium. In figure (119) we have represented a rigid, plain surface, with metric properties and, below, the shadow of its points.
(119)
A shadow has no proper shape, nor span. It depends on the screen and light rays production. On figure (120) we suggest the relativity of the shadow, with respect to the object.
(120)
These "parallel lines" are similar to those rays we introduced, to link the points of a sphere to its center. Here the points of the plane are "linked" to a "source" which is at infinity.
Give up this last idea of straight lines. Consider a bundle of cooked spaghetti (if they are not, they should be rigid and breakable). We can bend them. But we impose the spaghetti to keep joined together. Their neighborhood must not be modified.
(121)
All that is very crude, I know, and not fully rigorous. I just try to suggest to the reader what can be a manifold, a geometrical object without metric, where the main property is that any point has neighbors.
A manifold is a set of points m. I can imagine I associate any point of a manifold to a couple (M1, M2) of points, which belong to real surfaces, which own metric properties, length, and so on....
I call the n-dimensional manifold a skeleton-manifold and the associated n-dimensional surfaces just folds. Then I build the two-folds cover of a manifold.
On figure (122) the two-folds cover of a manifold m2 (two dimensions).
(122)
In figure (122) I have represented identical, parallel Euclidean folds (planes) with same metrics. But I can build the figure (123):
We will call M and M* conjugated points. Building these two folds from a "skeleton manifold" has a definite meaning: to any point M of the fold F we can associate one and only one conjugated point M*. There is a point to point mapping. Then we can forget the skeleton-manifold.
To the vicinity of any point of the fold F corresponds the vicinity of its conjugated point M*. See figure (124). This means that to any regular region of F corresponds a conjugated regular region that belongs to F*.
(124)
This shows in particular that the conjugated points M and M* are described by the same set of coordinates.