In differential geometry, a fibered manifold is surjective submersion of smooth manifolds Y → X. Locally trivial fibered manifolds are fiber bundles. Therefore, a notion of connection on fibered manifolds provides a general framework of a connection on fiber bundles.
Let π : Y → X be a fibered manifold. A generalized connection on Y is a section Γ : Y → J^{1}Y, where J^{1}Y is the jet manifold of Y.^{[1]}
With the above manifold π there is the following canonical short exact sequence of vector bundles over Y:

(1) 
where TY and TX are the tangent bundles of Y, respectively, VY is the vertical tangent bundle of Y, and Y ×_{X} TX is the pullback bundle of TX onto Y.
A connection on a fibered manifold Y → X is defined as a linear bundle morphism

(2) 
over Y which splits the exact sequence 1. A connection always exists.
Sometimes, this connection Γ is called the Ehresmann connection because it yields the horizontal distribution
of TY and its horizontal decomposition TY = VY ⊕ HY.
At the same time, by an Ehresmann connection also is meant the following construction. 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. Let
be two smooth paths in X and Y, respectively. Then t → y(t) is called the horizontal lift of x(t) if
A connection Γ is said to be the Ehresmann connection if, for each path x([0,1]) in X, there exists its horizontal lift through any point y ∈ π^{−1}(x([0,1])). A fibered manifold is a fiber bundle if and only if it admits such an Ehresmann connection.
Given a fibered manifold Y → X, let it be endowed with an atlas of fibered coordinates (x^{μ}, y^{i}), and let Γ be a connection on Y → X. It yields uniquely the horizontal tangentvalued oneform

(3) 
on Y which projects onto the canonical tangentvalued form (tautological oneform or solder form)
on X, and vice versa. With this form, the horizontal splitting 2 reads
In particular, the connection Γ in 3 yields the horizontal lift of any vector field τ = τ^{μ} ∂_{μ} on X to a projectable vector field
on Y.
The horizontal splitting 2 of the exact sequence 1 defines the corresponding splitting of the dual exact sequence
where T*Y and T*X are the cotangent bundles of Y, respectively, and V*Y → Y is the dual bundle to VY → Y, called the vertical cotangent bundle. This splitting is given by the verticalvalued form
which also represents a connection on a fibered manifold.
Treating a connection as a verticalvalued form, one comes to the following important construction. Given a fibered manifold Y → X, let f : X′ → X be a morphism and f ∗ Y → X′ the pullback bundle of Y by f. Then any connection Γ 3 on Y → X induces the pullback connection
on f ∗ Y → X′.
Let J^{1}Y be the jet manifold of sections of a fibered manifold Y → X, with coordinates (x^{μ}, y^{i}, y^{i}
_{μ}). 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 J^{1}Y → Y, and vice versa. It is an affine bundle modelled on a vector bundle

(4) 
There are the following corollaries of this fact.

(5) 
Given the connection Γ 3 on a fibered manifold Y → X, its curvature is defined as the Nijenhuis differential
This is a verticalvalued horizontal twoform on Y.
Given the connection Γ 3 and the soldering form σ 5, a torsion of Γ with respect to σ is defined as
Let π : P → M be a principal bundle with a structure Lie group G. A principal connection on P usually is described by a Lie algebravalued connection oneform on P. At the same time, a principal connection on P is a global section of the jet bundle J^{1}P → P which is equivariant with respect to the canonical right action of G in P. Therefore, it is represented by a global section of the quotient bundle C = J^{1}P/G → M, called the bundle of principal connections. It is an affine bundle modelled on the vector bundle VP/G → M whose typical fiber is the Lie algebra g of structure group G, and where G acts on by the adjoint representation. There is the canonical imbedding of C to the quotient bundle TP/G which also is called the bundle of principal connections.
Given a basis {e_{m}} for a Lie algebra of G, the fiber bundle C is endowed with bundle coordinates (x^{μ}, a^{m}
_{μ}), and its sections are represented by vectorvalued oneforms
where
are the familiar local connection forms on M.
Let us note that the jet bundle J^{1}C of C is a configuration space of Yang–Mills gauge theory. It admits the canonical decomposition
where
is called the strength form of a principal connection.