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Total Curvature.
...We can construct a sphere by juxtaposing mini posicones. However, during this operation, this surface with constant curvature (or curvature density, or local curvature) closes up. It therefore contains a certain amount of curvature, but how much?
...If I draw a geodesic triangle on a sphere, it encloses a certain number of mini posicones, a certain "quantity of curvature," which is an angle. This quantity is simply proportional to the area of the triangle, or more precisely, to the ratio between the area s of the triangle and the area S of the sphere.
...But we saw earlier that when we drew a geodesic triangle on a surface made from joined posicones, the deviation from the Euclidean sum was equal to the sum of the concentrated curvatures associated with each cone vertex contained within our triangle. Thus, it suffices to measure the sum of the angles a, b, and g of the triangle above, formed by three geodesic arcs of the sphere, to obtain a measure of the amount of angular curvature contained within this triangle. The geodesics of the sphere are its "great circles."
...Let us cut our sphere into eight equal portions. We will obtain eight triangles formed by geodesic arcs, whose three angles are right angles.
...Each of these triangles therefore contains a curvature equal to π/2. Since there are eight of them, the total curvature of the sphere is therefore 4π.
...This small observation serves to show that geometric results can be constructed using extremely simple reasoning.
...Returning to the theme of the blunted cone, we see that the flank of the object depends on the amount of curvature "contained inside," this curvature being either punctual (conical point) or distributed over a spherical cap. One can make the cap tend toward a point by reducing it homothetically (in such a way that it always contains the same "quantity of curvature").
Trajectories.
...In General Relativity, the key idea is simple: to assimilate the trajectories of objects, particles, photons, or matter to geodesics. Of course, these are geodesics of a four-dimensional hypersurface. Thus, we also have here only didactic images.
If we take our blunted cone, we can draw geodesics on it and project them onto a plane.
...All particles follow geodesics of the hypersurface: matter particles, but also photons and neutrinos. This is why we had fun depicting a geodesic that completely traverses the object. A neutrino can pass through the sun without any problem.
...But what is this plane onto which we project these geodesics? It is the way we represent space. Our "mental universe" is completely Euclidean and our thought is "flat." When we see a comet grazing the sun, it would never occur to us that in fact it is going "straight," that is to say, that it is following a geodesic of the hypersurface. Our perception of the world is Figure 24', where a celestial body "attracts" objects passing in its vicinity.
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