Construct allowing differentiation of tangent vector fields of manifolds
This article includes a list of general references, but it lacks sufficient corresponding inline citations. Please help to improve this article by introducing more precise citations.(February 2017) (Learn how and when to remove this message)
An affine connection on the sphere rolls the affine tangent plane from one point to another. As it does so, the point of contact traces out a curve in the plane: the development.
In differential geometry, an affine connection[a] 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. Connections are among the simplest methods of defining differentiation of the sections of vector bundles.[3]
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). The terminology is due to Cartan[b] and has its origins in the identification of tangent spaces in Euclidean space Rn 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 metric tensor then there is a natural choice of affine connection, called the Levi-Civita connection. The choice of an affine connection is equivalent to prescribing a way of differentiating vector fields which satisfies several reasonable properties (linearity and the Leibniz rule). This yields a possible definition of an affine connection as a covariant derivative or (linear) connection on the tangent bundle. A choice of affine connection is also equivalent to a notion of parallel transport, which is a method for transporting tangent vectors along curves. This also defines a parallel transport on the frame bundle. 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. 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.
^Lee 1997, p. 51.
^Lee 2018, p. 91.
^Lee 2018, p. 88, Connections.
^Akivis & Rosenfeld 1993, p. 213.
Cite error: There are <ref group=lower-alpha> tags or {{efn}} templates on this page, but the references will not show without a {{reflist|group=lower-alpha}} template or {{notelist}} template (see the help page).
In differential geometry, an affineconnection is a geometric object on a smooth manifold which connects nearby tangent spaces, so it permits tangent...
Look up affine in Wiktionary, the free dictionary. Affine may describe any of various topics concerned with connections or affinities. It may refer to:...
vanishing geodesic curvature. More generally, in the presence of an affineconnection, a geodesic is defined to be a curve whose tangent vectors remain...
geometry, the torsion tensor is a tensor that is associated to any affineconnection. The torsion tensor is bilinear map of two input vectors X , Y {\displaystyle...
mathematical physics, a spin connection is a connection on a spinor bundle. It is induced, in a canonical manner, from the affineconnection. It can also be regarded...
the affineconnection as the fundamental structure field rather than the metric tensor which was the original focus of general relativity. Affine connection...
of differential geometry, a Cartan connection is a flexible generalization of the notion of an affineconnection. It may also be regarded as a specialization...
affine transformation is an automorphism of an affine space (Euclidean spaces are specific affine spaces), that is, a function which maps an affine space...
array of numbers describing a metric connection. The metric connection is a specialization of the affineconnection to surfaces or other manifolds endowed...
framework) Connection (mathematics), a way of specifying a derivative of a geometrical object along a vector field on a manifold Connection (affine bundle)...
depends on the metric through the affineconnection. Whereas the covariant derivative required an affineconnection to allow comparison between vectors...
manifold is equipped with an affineconnection (a covariant derivative or connection on the tangent bundle), then this connection allows one to transport vectors...
assumed that the manifold carries its unique Levi-Civita connection. For a general affineconnection, the Ricci tensor need not be symmetric. Lott, John;...
In mathematics, affine geometry is what remains of Euclidean geometry when ignoring (mathematicians often say "forgetting") the metric notions of distance...
applied to real-world data. Affine holonomy groups are the groups arising as holonomies of torsion-free affineconnections; those which are not Riemannian...
In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent...
general relativity by removing a constraint of the symmetry of the affineconnection and regarding its antisymmetric part, the torsion tensor, as a dynamical...
a point p in a differentiable manifold equipped with a symmetric affineconnection are a local coordinate system in a neighborhood of p obtained by applying...
connection serves a similar purpose. More generally, differential geometers consider spaces with a vector bundle and an arbitrary affineconnection which...
metric-compatible affineconnection with Christoffel symbol Γ k i j {\displaystyle {\Gamma ^{k}}_{ij}} and the unique torsion-free Levi-Civita connection for the...
Transpose (2nd-order tensors) Related abstractions Affineconnection Basis Cartan formalism (physics) Connection form Covariance and contravariance of vectors...
In differential geometry, an affine manifold is a differentiable manifold equipped with a flat, torsion-free connection. Equivalently, it is a manifold...
approach given by a principal connection on the frame bundle – see affineconnection. In the special case of a manifold isometrically embedded into a higher-dimensional...
Global Information
