Connections on some functional bundles
We classify all -natural operators transforming second order connections Γ: Y → J²Y on a fibred manifold Y → M into second order connections on the vertical Weil bundle corresponding to a Weil algebra A.
For every product preserving bundle functor on fibered manifolds, we describe the underlying functor of any order . We define the bundle of -dimensional contact elements of the order on a fibered manifold and we characterize its elements geometrically. Then we study the bundle of general contact elements of type . We also determine all natural transformations of into itself and of into itself and we find all natural operators lifting projectable vector fields and horizontal one-forms...
In the framework of jet spaces endowed with a non-linear connection, the special curves of these spaces (h-paths, v-paths, stationary curves and geodesics) which extend the corresponding notions from Riemannian geometry are characterized. The main geometric objects and the paths are described and, in the case when the vertical metric is independent of fiber coordinates, the first two variations of energy and the extended Jacobi field equations are derived.