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 VE of E is the subbundle of the tangent bundle TE 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 TE which is complementary to VE, 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 VeE at e is the tangent space Te(Ex) to the fiber Ex through e. A horizontal bundle then determines an horizontal space HeE such that TeE is the direct sum of VeE and HeE.
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 = GLn.
Famous quotes containing the words horizontal and/or bundle:
“And yet out of eternity, a thread
separates itself on the blackness,
a horizontal thread
that fumes a little with pallor upon the dark.”
—D.H. (David Herbert)
“In the quilts I had found good objectshospitable, warm, with soft edges yet resistant, with boundaries yet suggesting a continuous safe expanse, a field that could be bundled, a bundle that could be unfurled, portable equipment, light, washable, long-lasting, colorful, versatile, functional and ornamental, private and universal, mine and thine.”
—Radka Donnell-Vogt, U.S. quiltmaker. As quoted in Lives and Works, by Lynn F. Miller and Sally S. Swenson (1981)