Horizontal Bundle

In mathematics, in the field of differential topology, given


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 eE with


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 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)