Affine connection

connectionaffineaffine connectionsconnectionsconnexioncurvaturecurvature tensorcurvature tensorslinear connectiontorsion free
In the branch of mathematics called differential geometry, an affine connection is a geometric object on a smooth manifold which connects nearby tangent spaces, so it permits tangent vector fields to be differentiated as if they were functions on the manifold with values in a fixed vector space.wikipedia
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Cartan connection

Cartan geometryconnectionsCartan
The notion of an affine connection has its roots in 19th-century geometry and tensor calculus, but was not fully developed until the early 1920s, by Élie Cartan (as part of his general theory of connections) and Hermann Weyl (who used the notion as a part of his foundations for general relativity). Infinitesimal parallel transport in the frame bundle yields another description of an affine connection, either as a Cartan connection for the affine group or as a principal connection on the frame bundle.
In the mathematical field of differential geometry, a Cartan connection is a flexible generalization of the notion of an affine connection.

Parallel transport

parallelparallel-transporttransported
A choice of affine connection is also equivalent to a notion of parallel transport, which is a method for transporting tangent vectors along curves.
If the manifold is equipped with an affine connection (a covariant derivative or connection on the tangent bundle), then this connection allows one to transport vectors of the manifold along curves so that they stay parallel with respect to the connection.

Covariant derivative

covariant differentiationtensor derivativecovariant differential
This yields a possible definition of an affine connection as a covariant derivative or (linear) connection on the tangent bundle.
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.

Levi-Civita connection

Christoffel symbolconnectionsLevi-Civita
If the manifold is further endowed with a Riemannian metric then there is a natural choice of affine connection, called the Levi-Civita connection.
More specifically, it is the torsion-free metric connection, i.e., the torsion-free connection on the tangent bundle (an affine connection) preserving a given (pseudo-)Riemannian metric.

General relativity

general theory of relativitygeneral relativity theoryrelativity
The notion of an affine connection has its roots in 19th-century geometry and tensor calculus, but was not fully developed until the early 1920s, by Élie Cartan (as part of his general theory of connections) and Hermann Weyl (who used the notion as a part of his foundations for general relativity).
:where s is a scalar parameter of motion (e.g. the proper time), and are Christoffel symbols (sometimes called the affine connection coefficients or Levi-Civita connection coefficients) which is symmetric in the two lower indices.

Differential geometry

differentialdifferential geometerdifferential geometry and topology
In the branch of mathematics called differential geometry, an affine connection is a geometric object on a smooth manifold which connects nearby tangent spaces, so it permits tangent vector fields to be differentiated as if they were functions on the manifold with values in a fixed vector space.
An important example is provided by affine connections.

Gauge covariant derivative

covariant derivative(gauge) covariant derivative
On the one hand, the ideas of Weyl were taken up by physicists in the form of gauge theory and gauge covariant derivatives.
Another approach is to understand the gauge covariant derivative as a kind of connection, and more specifically, an affine connection.

Development (differential geometry)

development
Differential geometers in the 19th century were interested in the notion of development in which one surface was rolled along another, without slipping or twisting.
From this point of view, rolling the tangent plane over a surface defines an affine connection on the surface (it provides an example of parallel transport along a curve), and a developable surface is one for which this connection is flat.

Lie derivative

Lie bracketLie commutatorcommuting vector fields
The torsion measures how closely the Lie bracket of vector fields can be recovered from the affine connection.
For a linear connection, the Lie derivative along X is :

Connection (vector bundle)

connectionKoszul connectionlinear connection
This yields a possible definition of an affine connection as a covariant derivative or (linear) connection on the tangent bundle.

Affine manifold

affine
In differential geometry, an affine manifold is a differentiable manifold equipped with a flat, torsion-free connection.

Connection (principal bundle)

principal connectionconnectionconnections
Infinitesimal parallel transport in the frame bundle yields another description of an affine connection, either as a Cartan connection for the affine group or as a principal connection on the frame bundle.
If the principal bundle P is the frame bundle, or (more generally) if it has a solder form, then the connection is an example of an affine connection, and the curvature is not the only invariant, since the additional structure of the solder form θ, which is an equivariant R n -valued 1-form on P, should be taken into account.

Torsion tensor

torsiontorsion-freetorsion form
The main invariants of an affine connection are its torsion and its curvature.
More generally, on a differentiable manifold equipped with an affine connection (that is, a connection in the tangent bundle), torsion and curvature form the two fundamental invariants of the connection.

Christoffel symbols

Christoffel symbolChristoffel coefficientsChristoffel connection
Correction terms were introduced by Elwin Bruno Christoffel (following ideas of Bernhard Riemann) in the 1870s so that the (corrected) derivative of one vector field along another transformed covariantly under coordinate transformations — these correction terms subsequently came to be known as Christoffel symbols.
The metric connection is a specialization of the affine connection to surfaces or other manifolds endowed with a metric, allowing distances to be measured on that surface.

Connection (mathematics)

connectionconnectionsconnected
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
Let M be a smooth manifold and let
Smooth closed manifolds have no local invariants (other than dimension), though geometric manifolds have local invariants, notably the curvature of a Riemannian manifold and the torsion of a manifold equipped with an affine connection.

Exponential map (Riemannian geometry)

exponential mapexponential map of this Riemannian metricExponential map, Riemannian geometry
This follows from the Picard–Lindelöf theorem, and allows for the definition of an exponential map associated to the affine connection.
. An affine connection on

Geodesic

geodesicsgeodesic flowgeodesic equation
Affine connections may also be used to define (affine) geodesics on a manifold, generalizing the straight lines of Euclidean space, although the geometry of those straight lines can be very different from usual Euclidean geometry; the main differences are encapsulated in the curvature of the connection.
More generally, in the presence of an affine connection, a geodesic is defined to be a curve whose tangent vectors remain parallel if they are transported along it.

Connection (affine bundle)

affine connections
. In particular, this is the case of an affine connection on the tangent bundle

Projective connection

complex projective structureprojective
Unparametrized geodesics are often studied from the point of view of projective connections.
The structure of a projective connection is modeled on the geometry of projective space, rather than the affine space corresponding to an affine connection.

Connection form

connectionconnection one-formvertical subspace
is such a smooth bilinear bundle homomorphism (called a connection form on M) then

Differentiable manifold

smooth manifoldsmoothdifferential manifold
In the branch of mathematics called differential geometry, an affine connection is a geometric object on a smooth manifold which connects nearby tangent spaces, so it permits tangent vector fields to be differentiated as if they were functions on the manifold with values in a fixed vector space.

Ehresmann connection

connectionEhresmannhorizontal lift
The principal connection α defines an Ehresmann connection on this bundle, hence a notion of parallel transport.
Note that (for historical reasons) the term linear when applied to connections, is sometimes used (like the word affine – see Affine connection) to refer to connections defined on the tangent bundle or frame bundle.

Derivative

differentiationdifferentiablefirst derivative
In the branch of mathematics called differential geometry, an affine connection is a geometric object on a smooth manifold which connects nearby tangent spaces, so it permits tangent vector fields to be differentiated as if they were functions on the manifold with values in a fixed vector space.

Vector space

vectorvector spacesvectors
In the branch of mathematics called differential geometry, an affine connection is a geometric object on a smooth manifold which connects nearby tangent spaces, so it permits tangent vector fields to be differentiated as if they were functions on the manifold with values in a fixed vector space.