  We propose in this paper a mathematicians' view of the Kaluza-Klein idea of a
five dimensional space-time unifying gravitation and electromagnetism, and
extension to higher-dimensional space-time. By considering the classification
of positive Einstein curvature tensors and the classical Cauchy-Choquet-Bruhat
theorems in general relativity, we introduce concepts of types and rigidity.
Then, 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, at least at this stage, 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 will then extend this setting to more
than 5 dimensions, giving a precise mathematical frame for possible additional
physical effects, preserving gravitation and electromagnetism.
