Horizontal bundle

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 V_eE at e is the tangent space T_e(E_x) to the fiber E_x through e. A horizontal bundle then determines a horizontal space H_eE such that T_eE is the direct sum of V_eE and H_eE.

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.

See also

• Horizontal form

References

• Kobayashi, Shoshichi and Nomizu, Katsumi (1996). Foundations of Differential Geometry, Vol. 1 (New ed.). Wiley Interscience. ISBN 0-471-15733-3.