Kaluza-Klein 5D Ideas Made Fully Geometric

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.