How can tangent vectors on a curved surface be uniquely combined? Let C be a curved surface and let P and Q be two points on C such that $P\neq Q$. Define four unique smooth curves, $γ_{1}(t)$, $γ_{2}(t)$, $γ_{3}(t)$ and $γ_{4}(t)$ on C, each curve joining P to Q along a different path. Let $T_{P}C$ be the tangent plane at P and let $v$ be a tangent vector $v\in T_{P}C$. Now, parallel transport $v$ along each curve from P to Q and let the resulting vectors be denoted $v_{γ_{1}}^{∥v}$, $v_{γ_{2}}^{∥v}$, $v_{γ_{3}}^{∥v}$, and $v_{γ_{4}}^{∥v}$. Since parallel transport is path dependent is it true that $v_{γ_{1}}^{∥v}=v_{γ_{2}}^{∥v}=v_{γ_{3}}^{∥v}=v_{γ_{4}}^{∥v}$, or equivalently, if $u=v_{γ_{1}}^{∥v}$+$v_{γ_{2}}^{∥v}$ and $w=v_{γ_{3}}^{∥v}$+$v_{γ_{4}}^{∥v}$, then $u=w$? If so, how can this be proven? If not, then can we conclude that there is no unique way to combine (add, subtract) vectors on a curved surface? In other words, how can vectors be uniquely combined on a curved surface?

In this image, tangent vectors at points $f_1,f_2,f_3$ and $f_4$ are parallel transported to a common point $P$ according to: $f_1 \rightarrow P, f_2 \rightarrow P, f_3 \rightarrow P$, and $f_4 \rightarrow P$ along the indicated counterclockwise paths (e.g along parallels and meridians). If the parallel transported vectors are then combined at $P$ (e.g. added), does the resulting vector have meaning in some way?
 A: 
In this reference, Parallel transport on a manifold, the author says "That is why we need a general definition of parallel transport, that allows us to compare different vectors at different points and gives us a prescription on how to move a vector around a manifold without really altering the vector." Statements like this are confusing me. Can we or can we not uniquely compare/combine vectors on a curved surface?

tl; dr: No, we cannot.

On a curved Riemannian manifold there is a natural concept of parallel transport along a piecewise-smooth path that allows tangent vectors to be "moved along the path," and therefore compared at different points of the path in a path-dependent sense.
Further, there are geometrically-special situations where parallel transport is (arguably) unique, e.g.:

*

*If the manifold is equipped with a metric of trivial holonomy so that parallel transport is path-independent, such as a Euclidean space, or a flat cylinder or torus;

*If any two points are joined by a unique minimizing geodesic, so there is a distinguished path for parallel transport.

Generally, however, there is no natural way to add or compare tangent vectors based at different points. I would not say parallel transport is performed "without really altering the vector." This animation loop shows why:

