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 .

Parallel transport

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.

  • 3
    $\begingroup$ If you have a length, then you have some sort of norm on the tangent space at every point, therefore a sort of Finsler metric. Since you mention angles between tangent vectors, this metric will be Riemannian. It is not true that any two points may be joined by a unique m.l.c. Think of the sphere and two opposite points: there are infinitely many m.l.c. connecting them. For Riemannian manifolds, completeness is a sufficient condition for any two points to be joined by a m.l.c., but not necessarily uniquely. It seems to me that what you want already exists in the framework of Riemannian gemetry. $\endgroup$ – Alex M. Jun 29 '18 at 12:50
  • $\begingroup$ Thanks for your answer, sorry I forget to write "Riemannian nanifold" not only "manifold". In case of completeness maybe the vector $w$ is the same regadless the m.l.c. used. In that case my construction could be possible. In textbooks I see always a parallel trasport that is given (or obtained by covariant derivative) but I have never seen a construction from scratch. $\endgroup$ – asv Jun 29 '18 at 13:25

Your assumption that minimal geodesics exist and are unique, in the complete case, is equivalent to $M$ being simply connected and having no conjugate points. This is already quite a restrictive class of manifolds.

In your above comment, you suggest that, in the case where multiple minimal geodesics between points exist, perhaps parallel transport is independent of the path taken. But that's typically not the case. For example, consider two antipodal points $p$ and $\tilde{p}$ on a two-dimensional sphere $S^2$. If you connect them via arcs of great circles $\gamma_1$ and $\gamma_2$ with initial vectors $v_1$ and $v_2$, respectively, that are perpendicular at $p$, then it's easy to see that parallel transporting $v_1$ around $\gamma_1$ and $\gamma_2$ produces vectors at $\tilde{p}$ that point in opposite directions.

Finally, while the angle between $V$ and $\gamma'$ is constant under parallel transport, that's not nearly restrictive enough to characterize it in dimension three or higher, because one can rotate around $\gamma'$.

  • $\begingroup$ Thank you for your answer. Regarding the constant angle we could take the tangent space of 2 dimension spanned by $V$ and $\gamma'$ and then all the curves who have tangent vector in that space. If I'm not wrong all these curves could make a 2 dimension manifold where $\gamma$ lives. $\endgroup$ – asv Jun 29 '18 at 20:57

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