  A general action is proposed for the fields of $q$-dimensional differential
form over the compact Riemannian manifold of arbitrary dimensions. Mathematical
tools come from the well-known de Rham-Kodaira decomposing theorem on the
harmonic integral. We have a field-theoretic action suitable for strings and
$p$-branes with or without spin degrees of freedom. In a completely-kinematical
way is derived the generalized Maxwell theory with a magnetic monopole over a
curved space-time, where we have a new type of gauge transformations.
