see mobies complex transformation

Where?

The question is set incorrectly. Parallel transport is inadequate operation in a geometry. It has been invented to remove the natural multivariance of a non-Euclidean geometry, when there are many vectors AB, AB1, AB2,...., which are equivalent to the vector CD, but vectors AB ,AB1, AB2,...are not equivalent between themselves.

Angles, when they exist, can be defined in terms of lengths, by the formulae of trigonometry, so length-preserving transformations (i.e. isometries) also preserve angles, e.g. in hyperbolic geometry. The converse is false, for instance Moebius transformations of the Riemann sphere (extended complex plane) are conformal (angle-preserving) but are not, in general, isometries. The concept of "parallel transport" needs to be defined for this question to be made more precise.

I mean parallel transport as define in Riemann geometry.

The parallel transport in Riemannian geometry is introduced, in order to make the Riemannian geometry single-variant. In single-variant geometry the relation of equality  of to vectors is transitive. Mathematican believe that in any geometry the equality relation must be transitive, because in any logical construction the equality relation must be transitive. Mathematicians make mistake at this point. They mixed geometry with .the method of the geometry construction. The conventional method of the geometry construction is a logical construction, but the geometry in itself  may   be not a logical construction. The Riemannian geometry is not a logical construction.

Euclidean method of the Euclidean geometry construction is a logical construction, when statments of a geometry are deduced from Euclidean axioms logically. However, exist another method of the geometry construction: deformation of the Euclidean geometry. All statements SE of Euclidean geometry (including dimension) are expressed in terms of Eucldean distance (metric) dE and only in trms of dE. Thereafter the Euclidean distance dE is replaced by the distance (metric) d  of other geometry G in all statements SE. As a resullt one obtains all statements S of the geometry G. The geometry G is multivariant, generally speaking. One cannot deduce the statements S from a systems of axoms logically, generally speaking.

The Riemannian geometry is a multivariant geometry. The parallel transport is not a true logical operation.

To talk about parallel transport (along curves) on a general manifold, you need a connection. On a manifold with a metric g (non-degenerate, so doesn't even have to be positive definite, can be Lorentzian as in General Relativity), parallel transport along any curve using the Levi-Civita connection preserves the metric g, so all inner products of vectors are preserved when you parallel transport them (along curves!). Any metric preserving connection would have the same property but the Levi-Civita connection is the unique torsion-free connection amongst all metric preserving connections and is the one most commonly used to compute geodesics, curvature, etc. If the curvature is zero and the manifold is simply connected, then parallel transport does not depend on the curve you choose, so you have "absolute parallelism" (the tangent bundle is trivialised by the flat connection).

I hope my answer helps, but you can find this explained better in any book on differential geometry.

Dear Maung Min-Oo, thanks for the answer. Your answer describes the geometry in the paper that I pointed out!

Dear Maung Min-Oo, you are quite right. The things, you have written, can be found in any textbook on differential geometry. But real problem is not a mathematical one. It is a physical problem: “To what extent can the Riemannian geometry  be applied for description of real space-time?” Constructing the general relativity, where the matter distribution influences on space-time geometry, one must use the most general class of space-time geometries. If one uses some subclass of possible space-time geometries, there is possible such a situation, when the real space-time geometry appears outside the considered subclass.

The mathematicians are convinced that the (pseudo)Riemannian geometries form the most general class of space-time geometries, because they knew only the proper Euclidean geometry and believe that any geometry is to be constructed by the same method as the Euclidean geometry. They cannot imagine that there is another method of the generalized geometries construction. There is another method of the space-time geometry construction. It is very simple. It is founded on a use of the proper Euclidean geometry GE, but not on the method of the geometry construction (as a logical deduction from axioms). All statements SE (including dimension) of the proper Euclidean geometry GE are presented in terms and only in terms of the Euclidean world function \sigmaE. Thereafter the Euclidean world function \sigmaE is replaced by the world function \sigma of a generalized geometry G in all statements SE of GE. As a result one obtains all statements S of G. This method of the geometry construction is called the deformation method. It does not contain any theorems (all necessary theorems are proved in GE). The obtained generalized geometry G is called the physical geometry. The set of Riemannian geometries is a small part of the set of physical geometries.

Physical geometries are multivariant in general. It means that vectors AB, AB1,…can be equal to some vector CD and vectors AB, AB1,…are not equivalent between themselves. The Riemannian geometry is multivariant in a natural way. But the conventional method of the geometry construction must not to lead to multivariance. The parallel transport is used to suppress the natural multivariance of the Riemannian geometry.

The deformation method of a space-time geometry construction is important in the relation, that the extended general relativity (EGR) constructed on the basis of physical geometries leads to other results, than the conventional general relativity (GR), constructed on the basis of Riemannian space-time geometries. In particular, in EGR there is no black holes, because the collapse is stopped by the arising antigravitation. In EGR a tachyon gas exists. It forms the dark matter. See papers

General relativity extended to non-Riemannian space-time geometry. Electronic Journal of Theoretical Physics 11, No. 31 (2014) pp 177-202, Available also at http://arXiv.org/abs/0910.3582v7

Induced antigravitation in the extended general relativity . Gravitation and Cosmology, , Vol. 18, No. 2, pp. 107– 112,( 2012).

Dynamic equations for tachyon gas, Int. J. Theor. Phys. 52, 133( 10), 3683- 3695, (2013)

In electronic form see on my site http://gasdyn-ipm.ipmnet.ru/~rylov/ialgr.htm

http://gasdyn-ipm.ipmnet.ru/~rylov/detg.htm

I have not calculated influence of  rotation. As to electromagnetic field, I do not think, that it will be influence on the antigraviation. The antigravitation effect appears as a result of influence of deviation from the Newtonian law of gravitation, it becomes to be effective only in the case, when the gravitational radius is of the order of the radius  of the heavy sphere. I do not see a connection between the EM field and the antigravitational effect.