The non-vanishing independent components of the christoffel symbol
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.
The components of the corresponding metric tensor h and the Christoffel symbols
on the manifold N will be denoted by [h.
Now, an alternative (although implicit) definition of the Christoffel symbols
is contained in the equation that states the vanishing of covariant derivatives of the metric:
In order to develop field equations, base vectors, metric coefficients and Christoffel symbols
are used in the curvilinear coordinates.
This theory has some advantages over the general relativity; the quantities such as christoffel symbols
and others become tensors which otherwise in Riemannian geometry they are not.
One straightforwardly goes through the tedious calculation of the Christoffel symbols
and the components of the Ricci tensor, finding:
n], n > k this leads to the Ricci tensor R, in components [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are Christoffel symbols
n]) is a coordinate chart on M, then the Christoffel symbols
v[lambda] are the Christoffel symbols
of the second kind, and [S.
CHRISTOFFEL SYMBOLS OF PSEUDO-RIEMANNIAN MANIFOLD (M,h), h = [[nabla]sup.
We recall that the Christoffel symbols
of second kind related to a given [THETA](4)-invariant spacetime metric  are the components of a [THETA](4)-invariant tensor field and depend on ten functions [B.
as the Christoffel symbols
(symmetric in their two lower indices) and