  A Lorentz-invariant cosmological model is constructed within the framework of
five-dimensional gravity. The five-dimensional theorem which is analogical to
the generalized Birkhoff theorem is proved, that corresponds to the Kaluza's
``cylinder condition''. The five-dimensional vacuum Einstein equations have an
integral of motion corresponding to this symmetry, the integral of motion is
similar to the mass function in general relativity (GR). Space closure with
respect to the extra dimensionality follows from the requirement of the absence
of a conical singularity. Thus, the Kaluza-Klein (KK) model is realized
dynamically as a Lorentz-invariant mode of five-dimensional general relativity.
After the dimensional reduction and conformal mapping the model is reduced to
the GR configuration. It contains a scalar field with a vanishing conformally
invariant energy-momentum tensor on the flat space-time background. This zero
mode can be interpreted as a vacuum configuration in GR. As a result the
vacuum-like configuration in GR can be considered as a manifestation of the
Lorentz-invariant empty five-dimensional space.
