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.