Show that getting parallel transported does not change angle between them

Question

I must tell you that I have never seen this kind of question in Tensor Analysis. Our professor had set up this question in our exam, but I don't know whether it belongs to Tensors or not. The question goes like this- If a vector $u^{i}$ gets parallel transported along a curve S, then $u^{i}; j$ $dX^{j}/dS=0$. If the angle between $u^{i}$ and $v^{j}$ is $\theta$, show that getting parallel transported does not change angle between them.

Answer 1

There is a certain point of view from which this is obvious. Parallel transport along a curve $C$ lying on a surface $S$ can be achived as follows, let $T$ be a developable surface which is tangent to $S$ along $C$, then $T$ can be mapped onto the plane. Now just move the vector along the surface $T$ in the ordinary planar sense. This gives one immediately that angles are preserved.