A connection on a fibered manifold Y → X is defined as a linear bundle morphism over Y which splits the exact sequence 1.
Any connection Γ on a fibered manifold Y → X yields a horizontal lift Γ ∘ τ of a vector field τ on X onto Y, but need not defines the similar lift of a path in X into Y.
This splitting is given by the vertical-valued form which also represents a connection on a fibered manifold.
Due to the canonical imbedding any connection Γ 3 on a fibered manifold Y → X is represented by a global section of the jet bundle J1Y → Y, and vice versa.
Given the connection Γ 3 on a fibered manifold Y → X, its curvature is defined as the Nijenhuis differential This is a vertical-valued horizontal two-form on Y.
It admits the canonical decomposition where is called the strength form of a principal connection.