# Connection (mathematics)

**connectionconnectionsconnectedConnectivityconnections in bundlesconnexionflat connectionssimply connected**

In geometry, the notion of a connection makes precise the idea of transporting data along a curve or family of curves in a parallel and consistent manner.wikipedia

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### Covariant derivative

**covariant differentiationtensor derivativecovariant differential**

An affine connection is typically given in the form of a covariant derivative, which gives a means for taking directional derivatives of vector fields, measuring the deviation of a vector field from being parallel in a given direction.

Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differential operator, to be contrasted with the approach given by a principal connection on the frame bundle – see affine connection.

### Holonomy

**holonomy groupholonomiesRiemannian holonomy**

The local theory concerns itself primarily with notions of parallel transport and holonomy.

In differential geometry, the holonomy of a connection on a smooth manifold is a general geometrical consequence of the curvature of the connection measuring the extent to which parallel transport around closed loops fails to preserve the geometrical data being transported.

### Ehresmann connection

**connectionEhresmannhorizontal lift**

An Ehresmann connection is a connection in a fibre bundle or a principal bundle by specifying the allowed directions of motion of the field.

In differential geometry, an Ehresmann connection (after the French mathematician Charles Ehresmann who first formalized this concept) is a version of the notion of a connection, which makes sense on any smooth fiber bundle.

### Parallel transport

**parallelparallel-transporttransported**

The local theory concerns itself primarily with notions of parallel transport and holonomy.

Other notions of connection come equipped with their own parallel transportation systems as well.

### Levi-Civita connection

**Christoffel symbolconnectionsLevi-Civita**

This latter means of parallel transport is the Levi-Civita connection on the sphere.

In Riemannian geometry, the Levi-Civita connection is a specific connection on the tangent bundle of a manifold.

### Differential geometry

**differentialdifferential geometerdifferential geometry and topology**

Differential geometry embraces several variations on the connection theme, which fall into two major groups: the infinitesimal and the local theory.

The apparatus of vector bundles, principal bundles, and connections on bundles plays an extraordinarily important role in modern differential geometry.

### Affine connection

**connectionaffineaffine connections**

For instance, an affine connection, the most elementary type of connection, gives a means for parallel transport of tangent vectors on a manifold from one point to another along a curve.

### Riemann curvature tensor

**Riemann tensorcurvature tensorcurvature**

Connections also lead to convenient formulations of geometric invariants, such as the curvature (see also curvature tensor and curvature form), and torsion tensor.

since the connection is torsionless, which means that the torsion tensor vanishes.

### Curvature

**curvednegative curvatureextrinsic curvature**

Connections also lead to convenient formulations of geometric invariants, such as the curvature (see also curvature tensor and curvature form), and torsion tensor.

### Shiing-Shen Chern

**S. S. ChernChernChen Xingshen**

Ehresmann connections were rather a solid framework for viewing the foundational work of other geometers of the time, such as Shiing-Shen Chern, who had already begun moving away from Cartan connections to study what might be called gauge connections.

He delivered his address on the Differential Geometry of Fiber Bundles. According to Hans Samelson, in the lecture Chern introduced the notion of a connection the principal fiber bundle, a generalization of the Levi-Civita connection.

### Élie Cartan

**CartanÉlie Joseph CartanE. Cartan**

As the twentieth century progressed, Élie Cartan developed a new notion of connection.

### Grothendieck connection

In algebraic geometry and synthetic differential geometry, a Grothendieck connection is a way of viewing connections in terms of descent data from infinitesimal neighbourhoods of the diagonal.

### Connection (fibred manifold)

**connectionconnection tangent-valued formconnections**

Therefore, a notion of connection on fibered manifolds provides a general framework of a connection on fiber bundles.

### Differential topology

**topologydifferentialtopology of manifolds**

Cartan connections were quite rigidly tied to the underlying differential topology of the manifold because of their relationship with Cartan's equivalence method.

Thus differential geometry may study differentiable manifolds equipped with a connection, a metric (which may be Riemannian, pseudo-Riemannian, or Finsler), a special sort of distribution (such as a CR structure), and so on.

### Connection (vector bundle)

**connectionKoszul connectionlinear connection**

A Koszul connection is a connection which defines directional derivative for sections of a vector bundle more general than the tangent bundle.

### Geometry

**geometricgeometricalgeometries**

In geometry, the notion of a connection makes precise the idea of transporting data along a curve or family of curves in a parallel and consistent manner.

### Tangent space

**tangent planetangenttangent vector**

For instance, an affine connection, the most elementary type of connection, gives a means for parallel transport of tangent vectors on a manifold from one point to another along a curve.

### Manifold

**manifoldsboundarymanifold with boundary**

For instance, an affine connection, the most elementary type of connection, gives a means for parallel transport of tangent vectors on a manifold from one point to another along a curve.

### Directional derivative

**normal derivativedirectionalderivative**

An affine connection is typically given in the form of a covariant derivative, which gives a means for taking directional derivatives of vector fields, measuring the deviation of a vector field from being parallel in a given direction.

### Vector field

**vector fieldsvectorgradient flow**

An affine connection is typically given in the form of a covariant derivative, which gives a means for taking directional derivatives of vector fields, measuring the deviation of a vector field from being parallel in a given direction.

### Derivative

**differentiationdifferentiablefirst derivative**

Thus a covariant derivative is a way of specifying a derivative of a vector field along another vector field on a manifold.

### Cartan connection

**Cartan geometryconnectionsCartan**

A Cartan connection is a way of formulating some aspects of connection theory using differential forms and Lie groups.