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 π : E → B {\displaystyle \pi \colon E\to B} , the vertical bundle V E {\displaystyle VE} and horizontal bundle H E {\displaystyle HE} are subbundles of the tangent bundle T E {\displaystyle TE} of E {\displaystyle E} whose Whitney sum satisfies V E ⊕ H E ≅ T E {\displaystyle VE\oplus HE\cong TE} . This means that, over each point e ∈ E {\displaystyle e\in E} , the fibers V e E {\displaystyle V_{e}E} and H e E {\displaystyle H_{e}E} form complementary subspaces of the tangent space T e E {\displaystyle T_{e}E} . 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 V e E {\displaystyle V_{e}E} at e ∈ E {\displaystyle e\in E} to be ker ( d π e ) {\displaystyle \ker(d\pi _{e})} . That is, the differential d π e : T e E → T b B {\displaystyle d\pi _{e}\colon T_{e}E\to T_{b}B} (where b = π ( e ) {\displaystyle b=\pi (e)} ) is a linear surjection whose kernel has the same dimension as the fibers of π {\displaystyle \pi } . If we write F = π − 1 ( b ) {\displaystyle F=\pi ^{-1}(b)} , then V e E {\displaystyle V_{e}E} consists of exactly the vectors in T e E {\displaystyle T_{e}E} which are also tangent to F {\displaystyle F} . 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 H e E {\displaystyle H_{e}E} of T e E {\displaystyle T_{e}E} is called a horizontal space if T e E {\displaystyle T_{e}E} is the direct sum of V e E {\displaystyle V_{e}E} and H e E {\displaystyle H_{e}E} . 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 H e E {\displaystyle H_{e}E} 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 ker ( d π e ) {\displaystyle \ker(d\pi _{e})} . 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 GL n {\displaystyle \operatorname {GL} _{n}} bundle.
Formal definition Let π:E→B be a smooth fiber bundle over a smooth manifold B. The vertical bundle is the kernel VE := ker(dπ) of the tangent map dπ : TE → TB. 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.
Example
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