Vertical and horizontal bundles
More precisely, given a smooth fiber bundle
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
) is a linear surjection whose kernel has the same dimension as the fibers of
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.
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.
The vertical bundle is the kernel VE := ker(dπ) of the tangent map dπ : TE → TB.
[2] Since dπe is surjective at each point e, it yields a regular subbundle of TE.
Furthermore, the vertical bundle VE is also integrable.
An Ehresmann connection on E is a choice of a complementary subbundle HE to VE in TE, called the horizontal bundle of the connection.
At each point e in E, the two subspaces form a direct sum, such that TeE = VeE ⊕ HeE.
The Möbius strip is a line bundle over the circle, and the circle can be pictured as the middle ring of the strip.
on the strip, the projection map projects it towards the middle ring, and the fiber is perpendicular to the middle ring.
is the tangent space to the fiber.
A simple example of a smooth fiber bundle is a Cartesian product of two manifolds.
Applying the definition in the paragraph above to find the vertical bundle, we consider first a point (m,n) in M × N. Then the image of this point under pr1 is m. The preimage of m under this same pr1 is {m} × N, so that T(m,n) ({m} × N) = {m} × TN.
The vertical bundle is then VB1 = M × TN, which is a subbundle of T(M ×N).
If we take the other projection pr2 : M × N → N : (x, y) → y to define the fiber bundle B2 := (M × N, pr2) then the vertical bundle will be VB2 = TM × N. In both cases, the product structure gives a natural choice of horizontal bundle, and hence an Ehresmann connection: the horizontal bundle of B1 is the vertical bundle of B2 and vice versa.
Various important tensors and differential forms from differential geometry take on specific properties on the vertical and horizontal bundles, or even can be defined in terms of them.
