Let $(M,g)$ be a Riemannian manifold and $p,q,r\in M$ points on the manifold. The vectors $u\in T_pM$, $v\in T_qM$ are then transported from $p$ resp. $q$ to $r$ along geodesics according to the parallel transport equation.
Suppose $\tilde{u}, \tilde{v}\in T_rM$ are those solutions. Can I assume these transported vectors are given in the same basis? If not, how can I transform them into the same base so I can e.g. add them? Is it even possible to formulate it in general?
Thanks in advance!
EDIT: I've come up with a work around, and maybe there isn't a simpler solution.
Suppose $\{e_i\}$ is an orthogonal basis of $T_rM$ and $\{e_j\}$ an OB of $T_pM$. We can parallel transport $\{e_i\}$ from $r$ to $p$ where it is an OB of $T_pM$, but in general $\{\tilde e_i\}\neq\{e_j\}$.
We can then write a vector in both bases $v=v^je_j=\tilde v^i\tilde e_i$, and calculate the components of the transported base as
$\tilde v^i=\dfrac{v^k}{\tilde e_ie^k}$
These components don't change under the parallel transport back from $p$ to $r$.