Vertical and horizontal bundles

In mathematics, the vertical bundle and the horizontal bundle are two subbundles of the tangent bundle of a smooth fiber bundle, forming complementary subspaces at each point of the fibre bundle. The vertical bundle consists of all vectors that are tangent to the fibers, while the horizontal bundle is then a particular choice of a subbundle of the tangent bundle which is complementary to the vertical bundle.
More precisely, if "π" : "E" → "M" is a smooth fiber bundle over a smooth manifold "M" and "e" ∈ "E" with "π"("e") = "x" ∈ "M", then the vertical space V"e""E" at "e" is the tangent space T"e"("E""x") to the fiber "E""x" containing "e". That is, "V""e""E" = T"e"(E"π"("e")). The vertical space is therefore a vector subspace of T"e""E". A horizontal space H"e""E" is then a choice of a subspace of T"e""E" such that T"e""E" is the direct sum of V"e""E" and H"e""E".
The disjoint union of the vertical spaces V"e""E" for each "e" in "E" is the subbundle V"E" of T"E": this is the vertical bundle of "E". Likewise, a horizontal bundle is the disjoint union of the horizontal subspaces H"e""E". The use of the words "the" and "a" in this definition is crucial: the vertical subspace is unique, it is determined solely by the fibration. By contrast, there are an infinite number of horizontal subspaces to choose from, in forming the direct sum.
The horizontal bundle concept is one way to formulate the notion of an Ehresmann connection on a fiber bundle. Thus, for example, if "E" is a principal "G"-bundle, then the horizontal bundle is usually required to be "G"-invariant: such a choice then becomes equivalent to the definition of a connection on the principal bundle. The choice of a "G"-invariant horizontal bundle and a connection are the same thing. In the case when "E" is the frame bundle, i.e., the set of all frames for the tangent spaces of the manifold, then the structure group "G" = GL"n" acts freely and transitively on each fibre, and the choice of a horizontal bundle gives a connection on the frame bundle.
Formal definition.
Let "π":"E"→"M" be a smooth fiber bundle over a smooth manifold "M". The vertical bundle is the kernel V"E" := ker(d"π") of the tangent map d"π" : T"E" → T"M".
Since dπe is surjective at each point "e", it yields a "regular" subbundle of T"E". Furthermore, the vertical bundle V"E" is also integrable.
An Ehresmann connection on "E" is a choice of a complementary subbundle H"E" to V"E" in T"E", called the horizontal bundle of the connection. At each point "e" in "E", the two subspaces form a direct sum, such that 
T"e""E" = V"e""E" ⊕ H"e""E".
Example.
A simple example of a smooth fiber bundle is a Cartesian product of two manifolds. Consider the bundle "B"1 := ("M" × "N", pr1) with bundle projection pr1 : "M" × "N" → "M" : ("x", "y") → "x". Applying the definition in the paragraph above to find the vertical bundle, we consider first a point (m,n) in "M" × "N". Then the image of this point under pr1 is m. The preimage of m under this same pr1 is {m} × "N", so that T(m,n) ({m} × "N") = {m} × T"N". The vertical bundle is then V"B"1 = "M" × T"N", which is a subbundle of T("M" ×"N"). If we take the other projection pr2 : "M" × "N" → "N" : ("x", "y") → "y" to define the fiber bundle "B"2 := ("M" × "N", pr2) then the vertical bundle will be V"B"2 = T"M" × "N".
In both cases, the product structure gives a natural choice of horizontal bundle, and hence an Ehresmann connection: the horizontal bundle of "B"1 is the vertical bundle of "B"2 and vice versa.
Properties.
Various important tensors and differential forms from differential geometry take on specific properties on the vertical and horizontal bundles, or even can be defined in terms of them. Some of these are: