For both physical and mathematical reasons, one expects a covariant derivative to be defined in terms of a limit. This is, however, a bit problematic. In three-dimensional Euclidean space limits can be defined uniquely in that vectors can be moved around without their lengths and directions changing, for instance, via the use of Cartesian coordinates (the set of basis vectors) and the usual dot product. Given these limits those corresponding to more general curvilinear coordinates can be established. The same is not true for curved spaces and/or spacetimes because they do not have an a priori notion of parallel transport.
Consider the classic example of a vector on the surface of a sphere (illustrated in Figure 2). Take that vector and move it along some great circle from the equator to the North pole in such a way as to always keep the vector pointing along the circle. Pick a different great circle, and without allowing the vector to rotate, by forcing it to maintain the same angle with the locally straight portion of the great circle that it happens to be on, move it back to the equator. Finally, move the vector in a similar way along the equator until it gets back to its starting point. The vector’s spatial orientation will be different from its original direction, and the difference is directly related to the particular path that the vector followed.
On the other hand, we could consider that the sphere is embedded in a three-dimensional Euclidean space, and that the two-dimensional vector on the sphere results from projection of a three-dimensional vector. Move the projection so that its higher-dimensional counterpart always maintains the same orientation with respect to its original direction in the embedding space. When the projection returns to its starting place it will have exactly the same orientation as it started with (see Figure 2). It is thus clear that a derivative operation that depends on comparing a vector at one point to that of a nearby point is not unique, because it depends on the choice of parallel transport.
Pauli  notes that Levi-Civita  is the first to have formulated the concept of parallel “displacement”, with Weyl  generalizing it to manifolds that do not have a metric. The point of view expounded in the books of Weyl and Pauli is that parallel transport is best defined as a mapping of the “totality of all vectors” that “originate” at one point of a manifold with the totality at another point. (In modern texts, one will find discussions based on fiber bundles.) Pauli points out that we cannot simply require equality of vector components as the mapping.
Let us examine the parallel transport of the force-free, point particle velocity in Euclidean three-dimensional space as a means for motivating the form of the mapping. As the velocity is constant, we know that the curve traced out by the particle will be a straight line. In fact, we can turn this around and say that the velocity parallel transports itself because the path traced out is a geodesic (i.e. the straightest possible curve allowed by Euclidean space). In our analysis we will borrow liberally from the excellent discussion of Lovelock and Rund . Their text is comprehensive yet readable for one who is not well-versed with differential geometry. Finally, we note that this analysis will be of use later when we obtain the Newtonian limit of the general relativistic equations, in an arbitrary coordinate basis.
We are all well aware that the points on the curve traced out by the particle can be described, in Cartesian coordinates, by three functions where is the standard Newtonian time. Likewise, we know that the tangent vector at each point of the curve is given by the velocity components , and that the force-free condition is equivalent to
What we need is a new “time” derivative , that yields a generally covariant statement
First note that the metric for our curvilinear coordinate system is obtainable from
We now identify our generally covariant derivative (dropping the overline) as
By relying on our understanding of the force-free point particle, we have built a notion of parallel transport that is consistent with our intuition based on equality of components in Cartesian coordinates. We can now expand this intuition and see how the vector components in a curvilinear coordinate system must change under an infinitesimal, parallel displacement from to . Setting Equation (28) to zero, and noting that , implies refers to the symbol as the “components of the affine relationship”, but we will use the modern terminology and call it the connection. In the language of Weyl and Pauli, this is the mapping that we were looking for.
For Euclidean space, we can verify that the metric satisfies
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