  After the 1916 success of General relativity that explained gravity by adding
time as a fourth dimension, physicists have been trying to explain other
physical fields by adding extra dimensions. In 1921, Kaluza and Klein has shown
that under certain conditions like cylindricity ($\partial g_{ij}/\partial
x^5=0$), the addition of the 5th dimension can explain the electromagnetic
field. The problem with this approach is that while the model itself is
geometric, conditions like cylindricity are not geometric. This problem was
partly solved by Einstein and Bergman who proposed, in their 1938 paper, that
the 5th dimension is compactified into a small circle $S^1$ so that in the
resulting cylindric 5D space-time $R^4\times S^1$ the dependence on $x^5$ is
not macroscopically noticeable. We show that if, in all definitions of vectors,
tensors, etc., we replace $R^4$ with $R^4\times S^1$, then conditions like
cylindricity automatically follow -- i.e., these conditions become fully
geometric.
