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- 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. (en)
- 미분기하학에서 수직 벡터 다발(垂直vector-, 영어: vertical vector bundle)은 올다발의 접다발 속의 특별한 부분 벡터 다발이다. 대략, 밑공간의 접다발을 "수평" 방향으로 간주하였을 때, 수직 벡터 다발은 순수하게 올 방향의, 즉 "수직" 방향의 벡터들로 구성된다. 반면, 올다발의 접다발 속의 "수평 벡터 다발"은 일반적으로 추가 구조 없이 정의되지 않는다. 이를 정의하기 위한 추가 구조는 에레스만 접속이라고 한다. (ko)
- 在数学微分几何领域，一个光滑纤维丛的铅直丛（vertical bundle）是切丛的一个，由所有和纤维相切的向量组成。更具体地，如果 π:E→M 是一个光滑流形 M 上一个光滑纤维丛，设 e ∈ E 满足 π(e)=x ∈ M，则在 e 处的铅直空间（vertical space） VeE 是纤维 Ex 包含 e 的切空间 Te(Ex)。这就是， VeE = Te(Eπ(e))。从而铅直空间是 TeE 的一个子空间，所有铅直空间的并是 TE 的一个子丛 VE，这便是 E 的铅直丛。 铅直丛是微分 dπ:TE→π-1TM 的核，这里 π-1TM 是拉回丛；用符号表示，VeE=ker(dπe)。因为 dπe 在每一点 e 是满射，它得出了商丛 TE/VE 与拉回 π-1TM 的一个典范等价。 E 上一个埃雷斯曼联络是选取 VE 在 TE 中的一个补子丛，称为这个联络的水平丛（horizontal bundle）。 (zh)
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- 미분기하학에서 수직 벡터 다발(垂直vector-, 영어: vertical vector bundle)은 올다발의 접다발 속의 특별한 부분 벡터 다발이다. 대략, 밑공간의 접다발을 "수평" 방향으로 간주하였을 때, 수직 벡터 다발은 순수하게 올 방향의, 즉 "수직" 방향의 벡터들로 구성된다. 반면, 올다발의 접다발 속의 "수평 벡터 다발"은 일반적으로 추가 구조 없이 정의되지 않는다. 이를 정의하기 위한 추가 구조는 에레스만 접속이라고 한다. (ko)
- 在数学微分几何领域，一个光滑纤维丛的铅直丛（vertical bundle）是切丛的一个，由所有和纤维相切的向量组成。更具体地，如果 π:E→M 是一个光滑流形 M 上一个光滑纤维丛，设 e ∈ E 满足 π(e)=x ∈ M，则在 e 处的铅直空间（vertical space） VeE 是纤维 Ex 包含 e 的切空间 Te(Ex)。这就是， VeE = Te(Eπ(e))。从而铅直空间是 TeE 的一个子空间，所有铅直空间的并是 TE 的一个子丛 VE，这便是 E 的铅直丛。 铅直丛是微分 dπ:TE→π-1TM 的核，这里 π-1TM 是拉回丛；用符号表示，VeE=ker(dπe)。因为 dπe 在每一点 e 是满射，它得出了商丛 TE/VE 与拉回 π-1TM 的一个典范等价。 E 上一个埃雷斯曼联络是选取 VE 在 TE 中的一个补子丛，称为这个联络的水平丛（horizontal bundle）。 (zh)
- 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. (en)
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- 수직 벡터 다발 (ko)
- Vertical and horizontal bundles (en)
- 铅直丛 (zh)
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