Vertical and horizontal bundles

vertical bundlehorizontalhorizontal bundlevertical tangent bundlehorizontal liftvertical vectorshorizontal distributionhorizontal one-formhorizontal subspacehorizontal'' and ''vertical'' vector fields
In mathematics, the vertical bundle and the horizontal bundle are two subbundles of the tangent bundle of a smooth fiber bundle, forming complementary subspaces at each point of the fibre bundle.wikipedia
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Ehresmann connection

connectionEhresmannhorizontal lift
The horizontal bundle concept is one way to formulate the notion of an Ehresmann connection on a fiber bundle.
: be the vertical bundle consisting of the vectors "tangent to the fibers" of E, i.e. the fiber of V at e\in E is.

Connection (principal bundle)

principal connectionconnectionconnections
Thus, for example, if E is a principal G-bundle, then the horizontal bundle is usually required to be G-invariant: such a choice then becomes equivalent to the definition of a connection on the principal bundle.
First note that the fundamental vector fields generating the G action on P provide a bundle isomorphism (covering the identity of P) from the bundle VP to, where VP = ker(dπ) is the kernel of the tangent mapping which is called the vertical bundle of P.

Exterior covariant derivative

covariant exterior derivativeexterior covariant differentiation
Suppose there is a connection on P; this yields a natural direct sum decomposition of each tangent space into the horizontal and vertical subspaces.

Tautological one-form

canonical one-formCanonical symplectic formLiouville form
The tautological one-form is the unique horizontal one-form that "cancels" a pullback.

Contorsion tensor

contorsion
This is an explicitly geometric viewpoint, with tensors now being geometric objects in the vertical and horizontal bundles of a fiber bundle, instead of being indexed algebraic objects defined only on the base space.

Mathematics

mathematicalmathmathematician
In mathematics, the vertical bundle and the horizontal bundle are two subbundles of the tangent bundle of a smooth fiber bundle, forming complementary subspaces at each point of the fibre bundle.

Subbundle

In mathematics, the vertical bundle and the horizontal bundle are two subbundles of the tangent bundle of a smooth fiber bundle, forming complementary subspaces at each point of the fibre bundle.

Tangent bundle

Canonical vector fieldrelative tangent bundletangent vector bundle
In mathematics, the vertical bundle and the horizontal bundle are two subbundles of the tangent bundle of a smooth fiber bundle, forming complementary subspaces at each point of the fibre bundle.

Fiber bundle

structure grouplocal trivializationtrivial bundle
The horizontal bundle concept is one way to formulate the notion of an Ehresmann connection on a fiber bundle. In mathematics, the vertical bundle and the horizontal bundle are two subbundles of the tangent bundle of a smooth fiber bundle, forming complementary subspaces at each point of the fibre bundle.

Differentiable manifold

smooth manifoldsmoothdifferential manifold
More precisely, if π : E → M is a smooth fiber bundle over a smooth manifold M and e ∈ E with π(e) = x ∈ M, then the vertical space V e E at e is the tangent space T e (E x ) to the fiber E x containing e.

Direct sum of modules

direct sumdirect sumsdirect sum of algebras
In mathematics, the vertical bundle and the horizontal bundle are two subbundles of the tangent bundle of a smooth fiber bundle, forming complementary subspaces at each point of the fibre bundle.

Disjoint union

disjointdisjoint union of sets
The disjoint union of the vertical spaces V e E for each e in E is the subbundle VE of TE: this is the vertical bundle of E.

Principal bundle

principalprincipal ''G''-bundleprincipal 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 then becomes equivalent to the definition of a connection on the principal bundle.

Frame bundle

orthonormal frame bundlelinear frame bundletangent frame bundle
In the case when E is the frame bundle, i.e., the set of all frames for the tangent spaces of the manifold, then the structure group G = GL n acts freely and transitively on each fibre, and the choice of a horizontal bundle gives a connection on the frame bundle.

Basis (linear algebra)

basisbasis vectorbases
In the case when E is the frame bundle, i.e., the set of all frames for the tangent spaces of the manifold, then the structure group G = GL n acts freely and transitively on each fibre, and the choice of a horizontal bundle gives a connection on the frame bundle.

General linear group

general lineargeneral linear Lie algebraGL
In the case when E is the frame bundle, i.e., the set of all frames for the tangent spaces of the manifold, then the structure group G = GL n acts freely and transitively on each fibre, and the choice of a horizontal bundle gives a connection on the frame bundle.

Group action (mathematics)

group actionactionorbit
In the case when E is the frame bundle, i.e., the set of all frames for the tangent spaces of the manifold, then the structure group G = GL n acts freely and transitively on each fibre, and the choice of a horizontal bundle gives a connection on the frame bundle.

Kernel (linear algebra)

kernelnull spacenullspace
The vertical bundle is the kernel VE := ker(dπ) of the tangent map dπ : TE → TM.

Pushforward (differential)

pushforwarddifferentialderivative
The vertical bundle is the kernel VE := ker(dπ) of the tangent map dπ : TE → TM.

Integrable system

integrableintegrable systemsintegrable model
Furthermore, the vertical bundle VE is also integrable.

Direct sum

direct summanddirect sum of abelian groupsdirect sums
At each point e in E, the two subspaces form a direct sum, such that

Cartesian product

productCartesian squareCartesian power
A simple example of a smooth fiber bundle is a Cartesian product of two manifolds.

Manifold

manifoldsboundarymanifold with boundary
A simple example of a smooth fiber bundle is a Cartesian product of two manifolds.

Tensor

tensorsorderclassical treatment of tensors
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.