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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 managed 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) keep the vectors and forget the sphere.
(117)
These half straight lines are infinite, but we figured
these 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 neighbours, which
form its neighbourhood. Each half straight line could be figured as enclosed
in a succession of cones ( figure (118) ) .
(118)
Around any line we can put as many lines of cones 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 coordinates 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 ) owns an infinite number
of neighbours ( x + Dx
; y + D y ) .
These "pre-metric" objects are called by mathematicians
manifolds .
It rather difficult to think about such "flexible" medium.
In (119) we have figured 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 infinite.
Give up this last idea of straight lines. Consider a bundle
of cooked spagheti ( if they are not, they should be rigid and breakable ).
We can bend it. But we impose the spagheti 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 neighbours.
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 (122) I have figured 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.
