Horizontal bundle

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 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".


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