  We are concerned with the relaxation and existence theories of a general
class of geometrical minimisation problems, with action integrals defined via
differential forms over fibre bundles. We find natural algebraic and analytic
conditions which give rise to a relaxation theory. Moreover, we propose the
notion of ``Riemannian quasiconvexity'' for cost functions whose variables are
differential forms on Riemannian manifolds, which extends the classical
quasiconvexity condition in the Euclidean settings. The existence of minimisers
under the Riemannian quasiconvexity condition has been established. This work
may serve as a tentative generalisation of the framework developed in the
recent paper: B. Dacorogna and W. Gangbo, Quasiconvexity and relaxation in
optimal transportation of closed differential forms, \textit{Arch. Ration.
Mech. Anal.} (2019), to appear. DOI:
\texttt{https://doi.org/10.1007/s00205-019-01390-9}.
