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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 in figure (125). Fix a string of length 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 points which lie at the same distance L from a given point S.

We can perform similar operations (figures (125)):

• On a horse saddle

• On a plane.
  (125)

On the plane surface, the perimeter is 2πL while the surface area of the disk is πL².

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 coincides with 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 than the equator's length.

Similar tests would show the negative curvature of the horse saddle. If we draw a closed curve, the set of points at the same distance L from a given point, on a horse saddle, the surface area of this negative curvature disk is larger than the surface area of a flat disk πL². Similarly, the perimeter of the negative curvature disk is larger than that of the flat disk: 2πL.

Geometry is a science for the blind. Geometers try to build tests that inhabitants of a given space could perform in order to discover by themselves its geometric properties. From the preceding figures, the inhabitants of a two-dimensional surface, unable to see this surface from an external point of view (for they live in it), could discover, through area and length measurements, if the portion of the surface they live in possesses 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 to 3D spaces. Choose a point O, somewhere. Take a string, a "pencil", and use it to draw the set of 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 constructed in a Euclidean 3D space, this area will be: 4πL².

If this area is found to be smaller, it means that this 3D space is not Euclidean. It is a Riemannian 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 points located at a given distance L from a fixed point O, will be larger than 4πL². 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 a presentation is limited. We must consider it as a crude didactic model.

The Riemann scalar curvature of an n-dimensional space is somewhat different.

In our present cosmological 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 the 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 the 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, are the null local curvature portions of the posicone and negacone.

Face to face, with point-to-point correspondence, are a constant positive curvature surface (a portion of a sphere) and a negative curvature surface (horse saddle). The curvature densities are equal and opposite. The circular borders are linked, point to point.

This is a didactic image of our cosmological 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.