  We study the real, massive Klein-Gordon field on a $C^\infty$
globally-hyperbolic background space-time with compact Cauchy hypersurfaces. In
particular, the parametrization of this system as initiated by Dirac and
Kucha\v{r} is put on a rigorous basis. The discussion is focussed on the
structure of the set of spacelike embeddings of the Cauchy manifold into the
space-time, and on the associated $e$-tensor density bundles and their tangent
and cotangent bundles. The dynamics of the field is expressed as a set of
automorphisms of the space of initial data in which each pair of embeddings
defines one such automorphism. Using these results, the extended phase space of
the system is shown to be a weak-symplectic manifold, and the Kucha\v{r}
constraint is shown to define a smooth constraint submanifold which is foliated
smoothly by the constraint orbits. The pull-back of the symplectic form to the
constraint surface is a presymplectic form which is singular on the tangent
spaces to the constraint orbits. Thus, the geometric structure of this
infinite-dimensional system is analogous to that of a finite-dimensional,
first-class parametrized system, and hence many of the results for the latter
can be transferred to the infinite-dimensional case without difficulty.
