However, this approach does not explain the geometry behind affine connections nor how they acquired their name.

Therefore, an affine connection is associated to a principal connection.

It is induced, in a canonical manner, from the affine connection.

An important example is provided by affine connections.

Let M be a manifold with an affine connection .

There are different physical interpretations of the translation part of affine connections.

In the modern approach, this is closely related to the definition of affine connections on the frame bundle.

An affine connection defines a notion of development of curves.

On any manifold of positive dimension there are infinitely many affine connections.

Affine connections can be defined within Cartan's general framework.

Curvature and torsion are the main invariants of an affine connection.

Affine connections from the point of view of Riemannian geometry.

They showed, however, that what appears to be a mass term involves the square of the affine connection .

Note there is a link between linear connection (also called affine connection) and a web.

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

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

A more mathematically motivated account of affine connections.

Like an affine connection, projective connections have associated torsion and curvature.

These are examples of affine connections.

Cartan's treatment of affine connections as motivated by the study of relativity theory.

Much like affine connections, though, projective connections also define geodesics.

Because an affine connection is not unique, it is an additional piece of data that must be specified on the manifold.

The complex history has led to the development of widely varying approaches to and generalizations of the affine connection concept.

The notion of an affine connection was introduced to remedy this problem by connecting nearby tangent spaces.

(Affine connections or affinities are more general terms used to encompass cases for which the space is not necessarily Riemannian).