  We consider a version of Kaluza-Klein theory where the cylinder condition is
not imposed. The metric is allowed to have explicit dependence on the "extra"
coordinate(s). This is the usual scenario in brane-world and space-time-matter
theories. We extend the usual discussion by considering five-dimensional
metrics with off-diagonal terms. We replace the condition of cylindricity by
the requirement that physics in four-dimensional space-time should remain
invariant under changes of coordinates in the five-dimensional bulk. This
invariance does not eliminate physical effects from the extra dimension but
separates them from spurious geometrical ones. We use the appropriate splitting
technique to construct the most general induced energy-momentum tensor,
compatible with the required invariance. It generalizes all previous results in
the literature. In addition, we find two four-vectors, J_{m}^{mu} and
J_{e}^{mu}, induced by off-diagonal metrics, that separately satisfy the usual
equation of continuity in 4D. These vectors appear as source-terms in equations
that closely resemble the ones of electromagnetism. These are Maxwell-like
equations for an antisymmetric tensor {F-hat}_{mu nu} that generalizes the
usual electromagnetic one. This generalization is not an assumption, but
follows naturally from the dimensional reduction. Thus, if {F-hat}_{mu nu}
could be identified with the electromagnetic tensor, then the theory would
predict the existence of classical magnetic charge and current. The splitting
formalism used allows us to construct 4D physical quantities from
five-dimensional ones, in a way that is independent on how we choose our
space-time coordinates from those of the bulk.
