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.[1] 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.

## . . . Vertical and horizontal bundles . . .

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*.[2]

Since dπ_{e} 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*.

A simple example of a smooth fiber bundle is a Cartesian product of two manifolds. Consider the bundle *B*_{1} := (*M* × *N*, pr_{1}) with bundle projection pr_{1} : *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 pr_{1} is m. The preimage of m under this same pr_{1} 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 pr_{2} : *M* × *N* → *N* : (*x*, *y*) → *y* to define the fiber bundle *B*_{2} := (*M*×*N*, pr_{2}) 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.

## . . . Vertical and horizontal bundles . . .

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