In mathematics, in the field of differential topology, given

*π*:*E*→*M*,

a smooth fiber bundle over a smooth manifold *M*, then the **vertical bundle** V*E* of *E* is the subbundle of the tangent bundle T*E* consisting of the vectors which are tangent to the fibers of *E* over *M*. A **horizontal bundle** is then a particular choice of a subbundle of T*E* which is complementary to V*E*, in other words provides a complementary subspace in each fiber.

In full generality, the horizontal bundle concept is one way to formulate the notion of an Ehresmann connection on a fiber bundle. However, the concept is usually applied in more specific contexts.

More precisely, if *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} through *e*. A horizontal bundle then determines an **horizontal space** H_{e}*E* such that T_{e}*E* is the direct sum of V_{e}*E* and H_{e}*E*.

If *E* is a principal *G*-bundle then the horizontal bundle is usually required to be *G*-invariant: see Connection (principal bundle) for further details. In particular, this is the case when *E* is the frame bundle, i.e., the set of all frames for the tangent spaces of the manifold, and *G* = GL_{n}.

### Famous quotes containing the words bundle and/or horizontal:

“We styled ourselves the Knights of the Umbrella and the *Bundle*; for, wherever we went ... the umbrella and the *bundle* went with us; for we wished to be ready to digress at any moment. We made it our home nowhere in particular, but everywhere where our umbrella and *bundle* were.”

—Henry David Thoreau (1817–1862)

“True. There is

a beautiful Jesus.

He is frozen to his bones like a chunk of beef.

How desperately he wanted to pull his arms in!

How desperately I touch his vertical and *horizontal* axes!

But I can’t. Need is not quite belief.”

—Anne Sexton (1928–1974)