Title: The fundamental Lepage form in variational theory for submanifolds,
Abstract: A setting for global variational geometry on Grassmann fibrations is
presented. The integral variational functionals for finite dimensional immersed
submanifolds are studied by means of the fundamental Lepage equivalent of a
homogeneous Lagrangian, which can be regarded as a generalization of the
well-known Hilbert form in the classical mechanics. Prolongations of
immersions, diffeomorphisms and vector fields to the Grassmann fibrations are
introduced as geometric tools for the variations of immersions. The first
infinitesimal variation formula together with its consequences, the
Euler-Lagrange equations for extremal submanifolds and the Noether theorem for
invariant variational functionals are proved. The theory is illustrated on the
variational functional for minimal submanifolds.