Specifically, I am interested in the case where the product of two three-dimensional spheres S3 × S3 evolves on a seven-dimensional sphere S7, because I believe (not without reason) that here it is necessary to look for the algebraic and geometric structure of space-time.
Dear Igor:
This is a very noble, pure-math question.
I admire it, but at the same time I am rather sure that it has almost nothing to do with physics.
Can you forgive me that statement?
Take care,
Otto
Otto, the connection with physics is still there. In fact, if the evolutionary movement of a globally minimal surface over a sphere is depicted as a static surface in a ball whose radius is equal to the evolutionary parameter (time), then in our case (when the product of spheres slides over a sphere) we can experiment with 8-dimensional space and surfaces of the level of a square of Euclidean and neutral metric forms. It turns out that the linear vector fields annulling the differential forms of these squares are isomorphic to the Dirac matrix algebra. Thus, with the help of vector fields, we can draw the coordinates of our space-time on the seven-dimensional sphere.
Dear Igor:
This is nice.
But I unfortunately found proof that the Einstein equation is faulty.
So the elegant methods have no use for physical reality if I am right.
Cordial wishes,
Otto
Dear Otto.
Of course, nature is multifaceted and does not have to always and everywhere obey certain equations (Maxwell, Einstein, etc.), but people will always try to describe it using equations. As for the inferiority of Einstein's equations, perhaps this is due to the fact that they are good only locally, and on a global scale do not adequately describe gravity.
Dear Igor:
i found that Einstein's ingenious founding example, the equivalence principle, has one more implication which he overlooked. It restores the (anyhow axiomatically presupposed) global constancy of c.
The consequences are maximally far-reaching and still largely unknown.
Thank you for this discussion,
Otto
I would like to draw your attention to another aspect of the problem of mathematical description of the evolution of globally minimal surfaces. A globally minimal surface can be considered as a foliation (or a typical foliation layer) and then the globally minimal surface will correspond to the globally minimal flow (orthogonal to this surface differential 1-form) that will evolve along with the foliation. If the minimal flows of the seven-dimensional sphere are compared with the flows of moving matter, then the Maxwell and Einstein equations will be only consequences of the minimal flow equation.
Most interesting mathematically.
Please, forgive me that in the aftermath of c-global 1929, this sophisticated class of mathematics spotted by you is most likely no longer of physical relevance -- in my momentary humble opinion.
Nevertheless your decisive help in this singular situation in scientific history is most cordially and even desperately invited.
May 15, 2019
Igor,
The general place to look is differential topology. Hopf vibrations are a classic tool when dealing with spheres.
The 7 sphere is a very interesting study with some unique properties and was extensively studied by Milnor and Bott. It is the first example of an exotic sphere.
Among other things the 7 sphere is parallelizable (the tangent bundle is trivial).
There are only three spheres that are parallelizable, S^1 (circle), S^3 since it is diffeomorphic to the Lie Group SU(2) and S^7. The Hairy Ball Theorem shows S^2 is not parallelizable. It can be shown that the product of two parallelizable manifolds is again parallelizable.
https://www.ams.org/journals/bull/1958-64-03/S0002-9904-1958-10166-4/S0002-9904-1958-10166-4.pdf
https://en.wikipedia.org/wiki/Exotic_sphere
http://math.uchicago.edu/~may/REU2015/REUPapers/McEnroe.pdf
https://ncatlab.org/nlab/show/7-sphere
Witten proposed that the 7 Sphere (and other Exotic spheres) can be interpreted as gradational instantons. See references here.
https://physics.stackexchange.com/questions/43683/7-sphere-is-there-any-physical-interpretation-of-exotic-spheres
In a recent thesis Nuchi considers embeddings of the Clifford Torus (S^3xS^3) into S7. That and the references there could be one place to start looking.
https://www.math.upenn.edu/grad/dissertations/NuchiThesis.pdf
Good luck
Truman,
Thanks for the interesting links. Many people are working in this direction and a lot of interesting things you can still learn.
I also draw your attention to the fact that the three-dimensional Euclidean observer space of our multidimensional mathematical model is well suited for measuring distances, since at the moment of evolution the diameter of one sphere is much larger than the diameter of another sphere of the product. However, due to the fact that the globally minimal surface has deviations from its shape, there are local changes in the diameters of the product, and hence the local deformation of the measured distances. As a result, taking into account the evolution of the product of spheres, the local observer "sees" a pseudo-Riemannian manifold.
Another interesting fact. If the evolution of the product of spheres (Clifford torus S3 × S3) is represented as the surface of the hypersphere of 8-dimensional space with a neutral metric having a radius equal to the evolutionary parameter, then we obtain a surface topologically equivalent to R4× S3 .
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