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].
Concretely, the metric tensor, the determinant of metric matrix field, the Christoffel symbols
, and Riemann tensors on the 3D domain are expressed by those on the 2D surface, which are featured by the asymptotic expressions with respect to the variable in the direction of thickness of the shell.
where the Christoffel symbols
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
Then the Christoffel symbols
[[nabla].sup.k.sub.j] of the Levi-Civita connection are related to the functions [g.sub.ij] by the formulas
where [omega] is the proper frequency of the photon, d[tau] is the interval of physically observable time, [c.sup.i] is the vector of the observable velocity of light ([c.sub.k][c.sup.k] = [c.sup.2]), [F.sub.i] is the gravitational inertial force, [A.sub.ik] is the angular velocity of the space rotation due to the non-holonomity of space (the non-orthogonality of the time lines to the spatial section), [D.sub.ik] is the deformation of space, [[DELTA].sup.i.sub.nk] are the three-dimensional Christoffel symbols
. Integration of the scalar equation should give a function E = E (t), where E = [??][omega] is the proper energy of the photon.