I have the following question. Assume we have a Riemannian manifold $M$ with the induced metric given by $d$.
I am looking for a canonical way to compare two elements $v,w\in T^n M$, where $T$ denotes the tangent bundle of a manifold.
For instance, for $n=0$, we can compare two elements $v,w\in M$ by $d(v,w)$.
For $n=1$ we can compare two elements $v,w\in TM$ by parallel transporting the vector $w$ to the basepoint $\pi(v)$ of $v$ and setting $D(v,w):= d(\pi(v),\pi(w)) + \|v-\tilde w \|$, where $\tilde w$ denotes the result of parallel transport of the vector $w$.
This is how far I get, for $n=2$ I cannot think of a natural way to compare two points. Is there a natural notion of distance on $T^nM$ for $n>1$?