Vertical and horizontal bundles

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the vertical bundle and the horizontal bundle are <a href="/facts/Vector_bundle/cXACAa1I">vector bundles</a> associated to a <a href="/facts/Fiber_bundle/SJ7Ek6cI">smooth fiber bundle</a>. 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 <a href="/facts/Subbundle/KFWrEleF">subbundles</a> of the <a href="/facts/Tangent_bundle/dq8zW9Kb">tangent bundle</a> 
 
 
 
 T
 E
 
 
 {\displaystyle TE}
 
 of 
 
 
 
 E
 
 
 {\displaystyle E}
 
 whose <a href="/facts/Whitney_sum/cXACAa1I">Whitney sum</a> 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 <a href="/facts/Complementary_subspace/Y6PBJnjg">complementary subspaces</a> of the <a href="/facts/Tangent_space/crvpbSeG">tangent space</a> 
 
 
 
 
 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 <a href="/facts/Direct_sum_of_vector_spaces/Y6PBJnjg">direct sum</a> of 
 
 
 
 
 V
 
 e
 
 
 E
 
 
 {\displaystyle V_{e}E}
 
 and 
 
 
 
 
 H
 
 e
 
 
 E
 
 
 {\displaystyle H_{e}E}
 
.
The <a href="/facts/Disjoint_union/x9QSfkr5">disjoint union</a> 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 <a href="/facts/Ehresmann_connection/qIpSLmFz">Ehresmann connection</a> on a <a href="/facts/Fiber_bundle/SJ7Ek6cI">fiber bundle</a>. Thus, for example, if E is a <a href="/facts/Principal_bundle/bbB2D94B">principal G-bundle</a>, then the horizontal bundle is usually required to be G-invariant: such a choice is equivalent to a <a href="/facts/Connection_(principal_bundle)/UchSqJ0C">connection on the principal bundle</a>. This notably occurs when E is the <a href="/facts/Frame_bundle/Gugt3qyL">frame bundle</a> associated to some vector bundle, which is a principal 
 
 
 
 
 GL
 
 n
 
 
 
 
 {\displaystyle \operatorname {GL} _{n}}
 
 bundle.

Vertical and horizontal bundles open-in-new

Vertical and horizontal bundles