I would know if it is possible to create a natural parallel transport on a Riemannian manifold $M$ with the following procedure.
Define a minimum length curve (m.l.c.) $\gamma$ from $P$ to $Q$ requiring to minimize the functional of length $L(\gamma)$ and show that for every two points $P$ and $Q$ in $M$ exists such m.l.c. and is unique .
Take two points $P$ and $Q$ in $M$ and a vector $v$ in $T_P M$ consider a m.l.c. $\gamma$ from $P$ to $Q$ and prove that there is a only vector field $V$ on $\gamma$ such that $V(P)=v$ and the angle between $V$ and $\gamma '$ is constant. Now we set $w=V(Q)$ as the parallel transport of $v$ in point $Q$.
I would know if the required proofs can be done (eventually in under some conditions) or if this procedure is in some textbook. I think that this porcedure is a natural way to construct a parallel trasport and maybe this parallel transport gives us also a Levi-Civita connection.