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cosmology of the twin universe Matter ghost matter astrophysics. 2: Conjugated steady state metrics. Exact solutions. (p5)
This operation can be extended to joined negacones (negative angular curvature density). For an eucliean surface C(M) = 0 everywhere. Using elementary negacones and small portions of a plane one can build any regular surface, where the angular curvature density C(M), positive, negative or zero, is a continuous function of the point M. We can now build a truncated posicone and join it to a portion of sphere. The continuity of the tangent plane is ensured if the angular curvatures q are equal. See figure 6.

Fig .6 : Building a smoothed "posicone".

A surface with constant negative angular curvature is called a horse saddle. See figure 7. On such a surface one can draw a curve centered on a point P.

Fig. 7 :** Building a "smoothed negacone".**

We can put a smoothed posicone and a smoothed negacone face to face, as shown on figure 1. Conjugated points M and M* have opposite curvature densities :
(61)

C(M*) = - C(M)

On the euclidean portions of the two conjugated surfaces these curvatures are zero :
(62)

C(M*) = C(M) = 0

We get an example of 2d conjugated geometries. Obviously, like in our 4d folds, the image of of a geodesic of a fold is definitively not a geodesic of the other one. See figures 8 and 9.

Fig. 8 : The image (composed by conjugated points) of a geodesic of the smoothed posicone F is not a geodesic of the smoothed negacone F.*

** ** Fig. 9 : The image (composed by conjugated points) of a geodesic of the smoothed negacone F is not a geodesic of the smoothed negacone F.* ** **
This is just a didactic image, but it illustrates the basic concept of conjugated geometries. In general relativity we deal with 4d hypersurfaces, whose metrics owns hyperbolic geometries, with signatures (+ - - -).

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