The non-vanishing independent components of the
christoffel symbol [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]
As an example, in the construction of GR, the role of
Christoffel symbol is obvious in describing both field and motion.
where [[GAMMA].sup.[rho].sub.[mu][nu]] is the
Christoffel symbols second kind and the square brackets mean
Following the cited book we call these functions the
Christoffel symbols of [nabla] with respect to the chart h and the local frame field [S.sub.U].
where the
Christoffel symbols [[GAMMA].sup.[lambda].sub.[mu]v] are
The components of the corresponding metric tensor h and the
Christoffel symbols on the manifold N will be denoted by [h.sub.[alpha][beta]],[H.sup.[alpha].sub.[beta][gamma]].
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:
If (U, [x.sup.1], ..., [x.sup.n]) is a coordinate chart on M, then the
Christoffel symbols [[GAMMA].sup.k.sub.ij] of the Levi-Civita connection are related to the functions [g.sub.ij] by the formulas
which is defined by the relation
