  In order to reduce the Klein-Gordon equation (with minimal coupling), we
introduce a generalization of the so-called "mode solutions" that are
well-known in the special case of a Robertson-Walker universe. After separation
of the variables, we end up with a partial differential equation in lower
dimension. A reduced version of the Gordon current arises and is conserved.
When the first factor-manifold is Lorentzian, distinct modes appear as mutually
orthogonal in the sense of the sesquilinear form obtained from the customary
Gordon current. Moreover, a sesquilinear form is defined on the space of
solutions to the reduced equation. Extension of this picture to curvature
coupling is possible when the second factor-manifold has a constant scalar
curvature.
