Around 1920, Kaluza and Klein had the idea to add a fifth dimension to the
classical 4-dimensional spacetime of general relativity to create a geometric
theory of gravitation and electromagnetism. Today, theoretical evidences, like
string theory, suggest the need for a spacetime with more than five dimensions.

We want to present in this paper a mathematical structure generalizing the
fiber bundle structure to the case of a product fiber of the form $F=S \times
W$, that enable the possible definition of multiple naturally defined fibers at
each point of the manifold, on which therefore one can define objects that
depend only on one of the components of the global fiber.

Although we do not pretend here to model precisely other known physical
interactions, we present 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 ; (we refer to the authors’ article [3]
). Also this geometry was a starting point for the second author’s new approach
to a geometric unification of General Relativity and Quantum Physics ( see

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