双宇宙宇宙学 物质-幽灵物质天体物理学。2： 共轭稳态度规。精确解。(p5)
此操作可以扩展到联结的负锥体（负的角曲率密度）。对于欧几里得表面，C(M) = 0 在任何地方都成立。使用基本的负锥体和小部分的平面，可以构建任何规则表面，其中角曲率密度 C(M)，正、负或零，是点 M 的连续函数。我们现在可以构建一个截断的正锥体，并将其连接到一个球面部分。如果角曲率 q 相等，就可以保证切平面的连续性。见图6。
图 .6 : 构建一个“平滑的正锥体”。
具有恒定负角曲率的表面称为马鞍面。见图7。在这样的表面上，可以画出一个以点 P 为中心的曲线。
图 .7 : 构建一个“平滑的负锥体”。
我们可以将一个平滑的正锥体和一个平滑的负锥体面对面放置，如图1所示。共轭点 M 和 M* 具有相反的曲率密度：
(61)
C(M*) = - C(M)
在两个共轭表面的欧几里得部分，这些曲率是零：
(62)
C(M*) = C(M) = 0
我们得到了一个二维共轭几何的例子。显然，就像在我们的四维折叠中一样，一个折叠的测地线的图像绝不是另一个的测地线。见图8和图9。
图 .8 : 平滑正锥体 F 的测地线的图像（由共轭点组成）不是平滑负锥体 F 的测地线。*
** ** 图 .9 : 平滑负锥体 F 的测地线的图像（由共轭点组成）不是平滑负锥体 F 的测地线。* ** **
这只是一个教学图像，但它说明了共轭几何的基本概念。在广义相对论中，我们处理的是四维超曲面，其度规具有双曲几何，符号为 (+ - - -)。
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原始版本（英文）
twin universe cosmology 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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