On the proper meaning of the curvature tensor and its general framework
We make evident a curvature tensor for every vector sub-bundle of an arbitrary manifold tangent bundle which reduces to the curvature tensor of an Ehresmann connection in the case of the horizontal sub-bundle of the tangent bundle to the total space of the nonlinear fiber bundle on which the connection is defined. Then the classical theorem of Frobenius would characterize the complete integrability of a vector sub-bundle of the tangent bundle by a zero curvature tensor in the sense of our definition here. A basic tool is a result about the curvature tensor of the natural lift of the vector sub-bundle to a manifold of maps with values in the base of that sub-bundle. Another is a localization property for a Lie algebra of vector fields over this manifold of maps.These allow to prove an additive formula for the curvature tensors of two supplementary sub-bundles. The main result consists in identifying a natural linear parallel transport on a supplementary vector sub-bundle along any tangent path to the vector sub-bundle under study, which is the right generalization of a linear connection parallel transport on a vector bundle along the projection in the base of that path. Then we derive the differential equation of the quotient of respective parallel transport operators induced by two different supplementary sub-bundles to the sub-bundle in question in terms of its curvature. Using this we obtain the equation of the infinitesimal variation of tangent paths to a vector sub-bundle, defined by its curvature, that appears as the root for the Jacobi equation of the infinitesimal variation of geodesics.