In this language, an affine connection is simply a covariant derivative or (linear) connection on the tangent bundle.However, this approach does not explain the geometry behind affine connections nor how they acquired their name.…

These last two points are quite hard to make precise, so affine connections are more often defined infinitesimally.

For the mentioned \alpha -families the affine connection is called the \alpha -connection and can also be expressed in more ways.

These locally defined connections are the same on overlapping parts, so there is a unique connection associated with an affine structure.Note there is a link between linear connection (also called affine connection) and a web.…

Although he initially assumed a symmetric affine connection, like Einstein he later considered the nonsymmetric field.

The complex history has led to the development of widely varying approaches to and generalizations of the affine connection concept.The most popular approach is probably the definition motivated by covariant derivatives.…

In particular, x t is a geodesic if and only if its development is an affinely parametrized straight line in T x 0 M.If M is a surface in R 3, it is easy to see that M has a natural affine connection.…

This is because it relies on the theorem on existence and uniqueness for ordinary differential equations which is local in nature.An affine connection is called complete if the exponential map is well-defined at every point of the tangent bundle.Intuitively speaking, the exponential map takes a given tangent vector to the manifold, runs along the geodesic starting at that point and going in that direction, for a unit time.…

Affine connections can be defined within Cartan's general framework.In the modern approach, this is closely related to the definition of affine connections on the frame bundle.Indeed, in one formulation, a Cartan connection is an absolute parallelism of a principal bundle satisfying suitable properties.…

Simplified derivation using an ancillary field instead of the usual affine connection.

The Einstein-Hilbert action for general relativity was first formulated purely in terms of the space-time metric.To take the metric and affine connection as independent variables in the action principle was first considered by Palatini.It is called a first order formulation as the variables to vary over involve only up to first derivatives in the action and so doesn't over complicate the Euler-Lagrange equations with terms coming from higher derivative terms.…

Affine connections can be defined within Cartan's general framework.In the modern approach, this is closely related to the definition of affine connections on the frame bundle.…

The main invariants of an affine connection are its torsion and its curvature.The torsion measures how closely the Lie bracket of vector fields can be recovered from the affine connection.…

The curvature of a spacetime can be characterised by taking a vector at some point and parallel transporting it along a curve on the spacetime.An affine connection is a rule which describes how to legitimately move a vector along a curve on the manifold without changing its direction.…

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.The main invariants of an affine connection are its torsion and its curvature.The torsion measures how closely the Lie bracket of vector fields can be recovered from the affine connection.…

Comparison of tangent vectors at different points on a manifold is generally not a well-defined process.An affine connection provides one way to remedy this using the notion of parallel transport, and indeed this can be used to give a definition of an affine connection.…

The notion of an affine connection was introduced to remedy this problem by connecting nearby tangent spaces.The origins of this idea can be traced back to two main sources: surface theory and tensor calculus.…

The terminology is due to Cartan and has its origins in the identification of tangent spaces in Euclidean space R n by translation: the idea is that a choice of affine connection makes a manifold look infinitesimally like Euclidean space not just smoothly, but as an affine space.On any manifold of positive dimension there are infinitely many affine connections.If the manifold is further endowed with a Riemannian metric then there is a natural choice of affine connection, called the Levi-Civita connection.…

The first three properties state that \nabla is an affine connection compatible with the metric, sometimes also called a hermitian or metric connection.

A useful way of measuring the curvature of a manifold is with an object called the Riemann (curvature) tensor.This tensor measures curvature by use of an affine connection by considering the effect of parallel transporting a vector between two points along two curves.The discrepancy between the results of these two parallel transport routes is essentially quantified by the Riemann tensor.…