# 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.

### Solder form

**canonical one-formsoldersoldering form**

### 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**

### 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**

### 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.