暗物质问题
（第5页）
。
5）
解的解释
从图2可以看出，势Y在无限远处趋于一个常数。在经典的Eddington解中，势具有对数增长。图3显示了一个位于区域s的物质团，被位于区域s*的平滑空洞包围。
在两个区域中，物质吸引物质。然而，来自场方程和泊松方程的负号使得物质和“对极物质”相互排斥。这有助于团的约束。对于给定的热速度，平衡压力所需的物质量因此减少。平滑的晕就像一个束腰。
场方程提供了宇宙的宏观描述。它不考虑物质的粒子性质。该模型表明，粒子和对极粒子存在于空间中非常遥远的对极区域。实际上，它们的性质是相同的。场方程的物理意义是：粒子和对极粒子通过引力相互作用，但不通过电磁力相互作用。我们假设对极粒子、团、环等无法通过望远镜或射电望远镜观测到。观测对极结构可能需要某种“引力望远镜”。
根据方程（22），团可以位于对极区域。因此，与之相关的大型晕，包围着广阔稀薄区域，也应该存在于可观测宇宙中。事实上，它们确实存在：这对应于我们所认为的宇宙大尺度结构：星系似乎围绕着巨大的稀薄气泡分布。根据我们的模型，对应区域中应该存在大量的对极物质云。
假设宇宙具有S3×R1的拓扑结构。读者可能很难理解这种奇特的三维几何结构。实际上，S3球体只是作为P3射影空间的双重覆盖来表示的。在这种配置中，球体上的每个点s都与它的对极点A(s)相关联。这种情况类似于S2球体覆盖P2射影空间，它在我们的三维空间R3中可以表示为著名的博伊表面。
图9：S2球体上的一个对极点对和博伊表面，即P2射影空间的图像
在图10中，我们描绘了球体的赤道及其在博伊表面上的位置。
图10：2-球体赤道附近及其在博伊表面的位置
。
图11显示了如何通过一个进行三次半扭转的莫比乌斯带将S2球体的赤道重新粘合到自身上。局部上，该表面可以被视为一个纤维化流形，其纤维具有两个值+1和-1。
图11：对应于莫比乌斯带覆盖的对映图像。
在3-球体S3中，如果沿着测地线前进，对极点位于中点。如果3-球体被嵌入四维空间中，那么任何一点都可以与其对极点重合。这些点对通过对极微分对合映射A相关联，但并不被识别。
如图12所示，我们可以从“奶酪”结构连续地过渡到团结构。这一特殊特征之前已通过二维数值模拟进行过说明。当空间的一个区域被放置在对极区域的“前面”，如图12所示，团会嵌入到“孔”中。
图12：宇宙大尺度整体结构的二维图像
图13：两个对极区域之间的相互作用
这种效应可能在星系结构层面起作用，如图14所示，每个星系嵌入到其共轭对极区域的“孔”中。
原始版本（英文）
The missing mass problem
(p5)
.
The interpretation of the solution
From the figure 2 we see that the potential
Ytends to a constant at the infinite. In the classical Eddington solution the potential owns a logarithmic growth. The figure 3 shows the association of a cluster of matter, located in the region
s
, sourrounded by a smooth hollow located in the region
s *.
In both regions matter attracts matter. But the negative sign, from the field equation and the Poisson equation, makes the matter and the "antipodal matter" to repel each other. This helps the confinement of the cluster. For a given thermal velocity the necessary quantity of matter to balance the pressure force is smaller. The smooth halo acts like a corset.
A field equation provides a macroscopic description of the universe. It does not take account of the corpuscular nature of matter. The model implies that particles and antipodal-particles live in very distant, antipodal portions of space. In fact their natures are identical. The physical meaning of the field equation is the following : the particules and antipodal-particles interact by gravitational effect, but not by electromagnetic effect. We assume that the antipodal particles, clusters, rings, are not observable with a telescope, or a radiotelescope. The observation of antipodal structures should require some sort of gravitational telescope.
From equation (22) clusters can be located in the antipodal region. Then, associated large halos, sourrounding wide rarefied regions, should exist in the observable universe too. In fact they do, for it corresponds, in our mind, to the observed large scale structure of the universe : the galaxies seem to be arranged around large rarefied bubbles. According to our model, large clouds of antipodal matter should exist in the corresponding associated antipodal regions.
The universe was assumed to have a S3
x
R1 topology. The reader has probabily some difficulties to understand this strange three-dimensional geometry. In fact the sphere S3 is simply shaped as the double cover of a projective space P3. In such arrangement each point
s
of the sphere is associated to its antipode A( s ). The situation is similar for a sphere S2 covering a projective space P2, which can be represented in our space R3 as the well known Boy surface.
Figure 9 : A couple of antipodal points on a sphere S2 and the Boy surface, image of the projective space P2
On the figure 10 we have figured the equator of a sphere and its location on the Boy Surface.
Figure 10 : The vicinity of the equator of a 2-sphere and its location on a Boy surface
.
The figure 11 shows how the equator of a S2 sphere can be glued on itself along a three half-turns Möbius belt. Locally the surface can be assimilated to a bundled manifold whose bundle owns two values + 1 and -1.
Figure 11 : Enantiomorphic image corresponding to the cover of a Möbius belt.
In a 3-sphere S3, if one follows a geodesic, the antipodal point is at the half-way. If the 3-sphere is immersed in a four-dimensional space it is possible to make any point and its antipode to coincidate. These couples of points are associated through the antipodal diffential involutive maping A, but not identified.
As shown on the figure 12 we can proceed continously from a "gruyère" structure to a cluster structure. This peculiar feature was illustrated before, through 2d numerical simulations. When a region of space is put "in front" of the antipodal region, as suggested in the figure 12, the clusters nest in the holes.
Figure 12 : Two-dimensional image of the global large structure of the universe
Figure 13 : The interaction between two antipodal regions
This effect could act at the level of the galactic structure, as suggested in the figure 14, each galaxy nesting in a "hole" of the conjugated antipodal region.