  This paper presents a generalization of symplectic geometry to a principal
bundle over the configuration space of a classical field. This bundle, the
vertically adapted linear frame bundle, is obtained by breaking the symmetry of
the full linear frame bundle of the field configuration space, and it inherits
a generalized symplectic structure from the full frame bundle. The geometric
structure of the vertically adapted frame bundle admits vector-valued field
observables and produces vector-valued Hamiltonian vector fields, from which we
can define a Poisson bracket on the field observables. We show that the linear
and affine multivelocity spaces and multiphase spaces for geometric field
theories are associated to the vertically adapted frame bundle. In addition,
the new geometry not only generalizes both the linear and the affine models of
multisymplectic geometry but also resolves fundamental problems found in both
multisymplectic models.
