We propose in this paper a new approach to the Kaluza-Klein idea of a five
dimensional space-time unifying gravitation and electromagnetism, and extension
to higher-dimensional space-time. By considering a natural geometric definition
of a matter fluid and abandoning the usual requirement of a Ricci-flat five
dimensional space-time, we show that a unified geometrical frame can be set for
gravitation and electromagnetism, giving, by projection on the classical
4-dimensional space-time, the known Einstein-Maxwell-Lorentz equations for
charged fluids. Thus, although not introducing new physics, we get a very
aesthetic presentation of classical physics in the spirit of general
relativity. The usual physical concepts, such as mass, energy, charge,
trajectory, Maxwell-Lorentz law, are shown to be only various aspects of the
geometry, for example curvature, of space-time considered as a Lorentzian
manifold; that is no physical objects are introduced in space-time, no laws are
given, everything is only geometry.

We then extend these ideas to more than 5 dimensions, by considering
spacetime as a generalization of a $(S^1\times W)$-fiber bundle, that we named
multi-fibers bundle, where $S^1$ is the circle and $W$ a compact manifold. We
will use this geometric structure as a possible way to model or encode
deviations from standard 4-dimensional General Relativity, or “dark” effects
such as dark matter or energy.

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