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Concept
Vertical and horizontal bundles
Summary
In mathematics, the vertical bundle and the horizontal bundle are vector bundles associated to a smooth fiber bundle. More precisely, given a smooth fiber bundle , the vertical bundle and horizontal bundle are subbundles of the tangent bundle of whose Whitney sum satisfies . This means that, over each point , the fibers and form complementary subspaces of the tangent space . The vertical bundle consists of all vectors that are tangent to the fibers, while the horizontal bundle requires some choice of complementary subbundle.
To make this precise, define the vertical space at to be . That is, the differential (where ) is a linear surjection whose kernel has the same dimension as the fibers of . If we write , then consists of exactly the vectors in which are also tangent to . The name is motivated by low-dimensional examples like the trivial line bundle over a circle, which is sometimes depicted as a vertical cylinder projecting to a horizontal circle. A subspace of is called a horizontal space if is the direct sum of and .
The disjoint union of the vertical spaces VeE for each e in E is the subbundle VE of TE; this is the vertical bundle of E. Likewise, provided the horizontal spaces vary smoothly with e, their disjoint union is a horizontal bundle. The use of the words "the" and "a" here is intentional: each vertical subspace is unique, defined explicitly by . Excluding trivial cases, there are an infinite number of horizontal subspaces at each point. Also note that arbitrary choices of horizontal space at each point will not, in general, form a smooth vector bundle; they must also vary in an appropriately smooth way.
The horizontal bundle is one way to formulate the notion of an Ehresmann connection on a fiber bundle. Thus, for example, if E is a principal G-bundle, then the horizontal bundle is usually required to be G-invariant: such a choice is equivalent to a connection on the principal bundle. This notably occurs when E is the frame bundle associated to some vector bundle, which is a principal bundle.
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