A Finsler geometry may be understood as a homogeneous variational problem, where the Finsler function is the Lagrangian. The extremals in Finsler geometry are curves, but in more general variational problems we might consider extremal submanifolds of dimension . In this minicourse we discuss these problems from a geometric point of view.
In the mid fifties, Charles Ehresmann defined Geometry as "the theory of more or less rich structures, in which algebraic and topological structures are generally intertwined". In 1973 he defined it as the theory of differentiable categories, their actions and their prolongations. Here we explain how he progressively formed this conception, from homogeneous spaces to locally homogeneous spaces, to fibre bundles and foliations, to a general notion of local structures, and to a new foundation of differential...
We present a generalization of the concept of semiholonomic jets within the framework of higher order prolongations of a fibred manifold. In this respect, a compilation of our 2-fibred manifold approach with the methods of natural operators theory is used.
In this work, we consider variational problems defined by -invariant Lagrangians on the -jet prolongation of a principal bundle , where is the structure group of . These problems can be also considered as defined on the associated bundle of the -th order connections. The correspondence between the Euler-Lagrange equations for these variational problems and conservation laws is discussed.
Given a Weil algebra and a smooth manifold , we prove that the set of kernels of regular -points of , , has a differentiable manifold structure and is a principal fiber bundle.
We review the approach to the calculus of variations using Ehresmann's theory of jets. We describe different types of jet manifold, different types of variational problem and different cohomological structures associated with such problems.
We classify all F2Mm1, m2, n1, n2-natural operators Atransforming projectable-projectable torsion-free classical linear connections ∇ on fibered-fibered manifolds Y of dimension (m1,m2, n1, n2) into rth order Lagrangians A(∇) on the fibered-fibered linear frame bundle Lfib-fib(Y) on Y. Moreover, we classify all F2Mm1, m2, n1, n2-natural operators B transforming projectable-projectable torsion-free classical linear connections ∇ on fiberedfibered manifolds Y of dimension (m1, m2, n1, n2) into Euler...