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Conjugated curvatures.****
How can we understand 3d spaces, with local positive or negative curvature ?
Start with 2d surfaces. Consider a sphere and fix a nail somewhere on it, as shown on figure (125). Fix a string, whose length is L, joining the nail and a pencil. We can use this to draw a circle, a *parallel *of the sphere. A parallel of a sphere is the set of the points which lie at the same distance L from a given point S.
We can do similar operations ( figures (125 ) :
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On a horse saddle
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On a plane.
(125)
On the plane surface the perimeter is 2 p L while the surface of the disk is p L2.
On the sphere the perimeter and the area of the disk are smaller. On the horse saddle, on the contrary, they are larger.
Consider a sphere and a parallel which identifies to its equator. See figure (126). The values correspond to figure (126)
(126)
The area of the disk is 3.875 times larger than the corresponding (grey) portion of the sphere. Its perimeter is 1.57 times longer that the equator's length
Similar tests would show the negative curvature of the horse saddle. If we draw a closed curve, set of points at the same distance L of a given point, on a horse saddle, the surface of this negative curvature disk is larger than the surface of a flat disk p L2. Similarly the perimeter of the negative curvature disk is larger than the one of the flat disk : 2 p L .
Geometry is a science for the blind. Geometers try to build tests that inabitants of a given space could operate in order to discover by themselve its geometric properties. From the precedent figures, the inhabitants of a two dimensional surface, unable to see this one from an external point of view (for the live in) could discover, through area and length measurement, if the portion of a surface they live in owns a positive local curvature, a negative local curvature, or a null local curvature (euclidean space).
Notice that there are surfaces whose local curvature can be positive, null or negative. Example : a torus.
(126ter)
Similar methods apply for 3d spaces. Choose a point O , somewhere. Take a string, a "pencil", and use it to draw the set of the points located at a given distance L from the considered point. You get a sphere and you can measure its area. If this surface has been built in an euclidean 3d space, this area will be : 4 p L2 .
If this area is found smaller, it means that this 3d space is not euclidean. It is a riemanian 3d space, with positive curvature. If we measure the volume we will find it smaller than :
(127)
The situation will be reversed if we deal with a negatively curved 3d space. The area of the sphere, considered as the set of point located at a given distance L of a fixed point O will be larger than 4 p L2 . The volume, inside that closed surface, will be larger than (127).
Cosmology is not founded on simple 3d spaces, but on 4d-hypersurfaces ( with "hyperbolic signature" ) so that such presentation is limited. We must consider it as a crude didactic model.
The Riemann, scalar curvature of a n-dimensional space is somewhat different.
In our present cosmologic model, we will assume that the local scalar riemann curvature, at conjugated points ( M , M ) are opposite :
*(127bis)
R* = - R
The specialist will find more details in the paper :
J.P.Petit & P.Midy : Matter ghost matter astrophysics. 2 : Conjugated steady state metrics. Exact solutions. Geometrical Physics A , 5 , march 1998.
Next, a useful 2d didactic image, which corresponds to figure 39.
(128)
Top : a smoothed posicone. Null local (angular) curvature in the portion of posicone. Constant positive (angular) curvature density in the (grey) portion of a sphere.
Below : a "smoothed negacone". Null local (angular) curvature density in the portion of negacone surrounding the horse saddle. Constant negative (angular) curvature density in the portion of the horse saddle, facing the portion of a sphere.
The curvatures are conjugated. Face to face, with point to point correspondence, null local curvature portions of posicone and negacone.
Face to face, with point to point correspondence, a constant positive curvature surface (a portion of a sphere) and a negative curvature surface (horse saddle). The curvature densities are equal end opposite. The circular borders are linked, point to point.
This is a didactic image of our cosmologic model. For more mathematical details, see :
J.P.Petit & P.Midy : Matter ghost-matter astrophysics. 1.The geometrical framework. The matter era and the newtonian approximation. Geometrical Physics A, 4 , march 1998.