affine connection


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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.
Then the affine connection in (11) for the [mu] = 1 component reduces to
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;
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;
Thus, we construct the explicit expressions for the coefficients of affine connection in this gravitational field 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
6) can also be obtained by constructing the coefficients of affine connection for this gravitational field and evaluating the time equation of motion for particles of non-zero rest masses.