  In a 5-dimensional spacetime ($M,g_{ab}$) with a Killing vector field $\xi
^a$ which is either everywhere timelike or everywhere spacelike, the collection
of all trajectories of $\xi ^a$ gives a 4-dimensional space $S$. The reduction
of ($M,g_{ab}$) is studied in the geometric language, which is a generalization
of Geroch's method for the reduction of 4-dimensional spacetime. A
4-dimensional gravity coupled to a vector field and a scalar field on $S$ is
obtained by the reduction of vacuum Einstein's equations on $M$, which gives
also an alternative description of the 5-dimensional Kaluza-Klein theory.
Besides the symmetry-reduced action from the Hilbert action on $M$, an
alternative action of the fields on $S$ is also obtained, the variations of
which lead to the same fields equations as those reduced from the vacuum
Einstein equation on $M$.
