affine connection


Also found in: Wikipedia.

affine connection

[ə′fīn kə′nek·shən]
(mathematics)
A structure on an n-dimensional space that, for any pair of neighboring points P and Q, specifies a rule whereby a definite vector at Q is associated with each vector at P ; the two vectors are said to be parallel.
References in periodicals archive ?
Golab [1] introduced and studied quarter symmetric connection in a Riemannian manifold with an affine connection, which generalizes the idea of semi symmetric connection.
His topics include the basics of geometry and relativity, affine connection and covariant derivative, the geodesic equation and its applications, curvature tensor and Einstein's equation, black holes, and cosmological models and the big bang theory.
The affine connection vanishes when there is no gravitational distortion; so for the point mass m, it should be solely a function of the curvature distortion [n.
The affine connection can be related to the the metric coefficients [g.
Stick with the Lorentz group SO(1, 3) or introduce an affine structure in the fibres (to be sharply distinguished from an affine connection on the bundle), so the local symmetry group becomes the inhomogeneous Lorentz group, that is, the Poincare group.
From this covariant metric tensor, we can then construct our field equations for the gravitational field after formulating the Coefficients of affine connection, Riemann Christoffel tensor, Ricci tensor and the Einstein tensor [7-12].
or more explicitly interms of the affine connections, Ricci tensor and covariant metric tensor as;
These vectors are defined with the help of the affine connection (or inertial structure) of the spacetime, and this connection may be partially or wholly determined by the overall material (stress-energy) distribution via the field equations of GTR.
It is well known that the coefficients of affine connection for any gravitational field are defined in terms of the metric tensor [14, 15] as;
It can be shown that the coefficients of affine connection for the gravitational field exterior to a homogenous oblate spheroidal mass are given in terms of the metric tensors for the gravitational field as
Among the topics are the past and future of differential geometry, multi-linear algebra, exterior differential forms, affine connections, review of surface theory, and the pseudo-spherical surface.