My question is in response to Dirac's assertion:
Suppose we have a vector $\left\{ A^{\mu}\right\}$ at a point $\mathscr{P}$. If the space is curved, we cannot give meaning to a parallel vector at a distant point $\mathscr{Q}$, as one can easily see if one thinks of the example of a curved two-dimensional space in a three-dimensional Euclidean space.
The statement is from Chapter 6 of General Theory of Relativity by P.A.M Dirac.
Let us restrict consideration to the surface of a globe such as an idealized Earth surface. We shall further restrict consideration to an open "nice" region which is small relative to the radius of the globe. To be specific, let us describe a boundary $\partial{\mathcal{D}_{o}}$ which is at a constant geodesic distance of 1000 kilometers from some designated location $\mathscr{P}_{o}$. We are assuming our globe is measured with units proportional in scale to the size of Earth. We call $\mathcal{D}_{o}$ the open region bounded by $\partial{\mathcal{D}_{o}}$ with the smaller area.
Suppose now that we have a disk $\delta\mathcal{D}_{o}$ of geodesic radius 1 meter, centered on $\mathscr{P}_{o}$, upon which we draw two perpendicular two-space diameters (which intersect at $\mathscr{P}_{o}$). Since $\delta\mathcal{D}_{o}$ is practically indistinguishable form a disk living in the tangent Euclidean plane, using naive concepts of positively directed segments, we designate one of the (half-diameter) radial segments $\mathfrak{\hat{e}}_{x}$, and another $\mathfrak{\hat{e}}_{y}$. These constitute an orthonormal spanning basis of the tangent plane at $\mathscr{P}_{o}$.
It seems intuitively obvious to me that this disk $,\delta\mathcal{D}_{o}$ can be parallel transported along geodesics intersecting at $\mathscr{P}_{o}$ to every point $\mathscr{Q}$ in $\mathcal{D}_{o}$ so that, at every point $\mathscr{Q}$ in $\mathcal{D}_{o}$ there is uniquely determined image of $\delta\mathcal{D}_{o}$.
My question is this: in my restricted example, have I not provided a definition of "parallel" such that at every point $\mathscr{Q}\in\mathcal{D}_{o}$ there is a one-to-one mapping between the tangent plane $\mathcal{T}_{\mathscr{Q}}$ and $\mathcal{D}_{o}$?
I am not pretending to have refuted Dirac's assertion. I am merely attempting to determine its exact meaning.