Addition of vectors given at different points on a manifold Let's say I have a set of tangent vectors given at different points ($\vec{v}_i \in T_{p_i}(M)$) on a riemannian manifold with metric and compatible levi civita connection and I like to calculate e.g. the "mean" of all vectors of this set. So I need to add vectors in some sense.
Now I have learned that without connection addition of vectors does not make sense since they are all defined in different tangent spaces.
How does the concept of parallel transport fix ($\vec{v}_1,p_1$) + ($\vec{v}_2,p_2$) ?
Do I understand this correctly that I define a vector addition at one point e.g. $p_1$ ($\vec{v}_1,p_1$) + ($\vec{v}_2,p_2$) as:
Find the geodesic between $p_1$ and $p_2$ with tangent vector $\vec{v}_2$ at $p_2$ (solve this set of ODEs) and calculate the tangent vector $\vec{\hat{v}}_2$ of this curve at $p_1$ so that  at $p_1$ ($\vec{v}_1,p_1$) + ($\vec{v}_2,p_2$) := ($\vec{v}_1,p_1$) + ($\vec{\hat{v}}_2,p_1$) = ($\vec{v}_1+\vec{\hat{v}}_2,p_1$) 
Is this the correct conceptual approach or do I miss a property of the connection which makes it obvious how this should work now?
 A: The parallel transport mechanism allows you to identify tangent planes at different points $p, q \in M$ only when you specify in addition a curve $\gamma \colon [0,1] \rightarrow M$ connecting $p$ and $q$ (so that $\gamma(0) = p, \gamma(1) = q$). Then you get a linear isomorphism $P = P_{\gamma,0,1} \colon T_pM \rightarrow T_q M$ and given $v \in T_pM, w \in T_qM$, you can use $P$ to define the addition of $v$ and $w$ in $T_pM$ as $v + P^{-1}(w)$. In general, the map $P$ strongly depends on the path $\gamma$ and choosing different paths will result in different linear isomorphisms.
However, usually one uses this identification only for "infinitesimally near" points and in this case, it does not depend on the path one uses to identify the tangent planes. For example, if you will define the derivative of a vector field $X$ with respect to a direcion $v \in T_pM$ at a point $p$ by
$$ ((\nabla_{v})(X))(p) = "\lim_{t \to 0} \frac{X(\gamma(t)) - X(p)}{t}" = \lim_{t \to 0} \frac{P_{\gamma,t,0}^{-1}(X(\gamma(t)) - X(p)}{t} = \frac{d}{dt} P_{\gamma,t,0}^{-1}(\gamma(t))|_{t=0} $$
where $\gamma \colon (-\varepsilon, \varepsilon) \rightarrow M$ is a path with $\gamma(0) = p, \dot{\gamma}(0) = v$ then the result won't dependent on the choice of the path $\gamma$ and will result in the notion of a covariant derivative of a vector field. The expression $P_{\gamma,t,0}^{-1}(\gamma(t)) - \gamma(0)$ will depend on the choice of the path $\gamma$ but its derivative at $t = 0$ will not.
If you are working with a Riemannian metric, then given $p, q \in M$ with $q$ close enough to $p$, there is a unique geodesic $\gamma$ connecting $p$ to $q$ and you can use this "more canonical" choice of path in order to identify $T_pM$ and $T_qM$ but this only works with points $p,q$ that are close to each other.
