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Some cohomological aspects of the calculus of variations on finite-order jets of submanifolds

We show that on finite-order jets of submanifolds there exists a sequence, the finite-order variational sequence, which allows to formulate the calculus of variations in a purely differential-geometric context. We provide a natural representation of all elements of the above sequence. This allows us to develop a calculus of variations on jets of arbitrary order, which generalizes the constructions of Bryant, Griffiths and Grossman [8]. We obtain a new classification of null Lagrangians of any specified order and minimal order Lagrangians for variationally trivial Euler–Lagrange morphisms. As an example we provide an intrinsic formulation of the equation of minimal submanifolds of a Riemannian manifold as a Euler–Lagrange equation. MSC 2000 classification: 58A20 (primary), 53A10, 58A12, 58E99, 58J10.