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