## affine connection

73 examples (0.01 sec)
• 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.
• 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.
• If M is a surface in R 3, it is easy to see that M has a natural affine connection.
• An affine connection is called complete if the exponential map is well-defined at every point of the tangent bundle.
• In the modern approach, this is closely related to the definition of affine connections on the frame bundle.
• Simplified derivation using an ancillary field instead of the usual affine connection.
• To take the metric and affine connection as independent variables in the action principle was first considered by Palatini.
• Affine connections can be defined within Cartan's general framework.
• The torsion measures how closely the Lie bracket of vector fields can be recovered from the affine connection.
• An affine connection is a rule which describes how to legitimately move a vector along a curve on the manifold without changing its direction.
• The main invariants of an affine connection are its torsion and its curvature.
• 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.
• On any manifold of positive dimension there are infinitely many affine connections.
• The first three properties state that \nabla is an affine connection compatible with the metric, sometimes also called a hermitian or metric connection.
• 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.