parallel transport composition

Question

Is it correct that $$P_b^cP_a^bv = P_a^cv,$$ where $P_a^b$ denote parallel transport operator from $T_a\mathcal{M}$ to $T_b\mathcal{M}$ and $v\in T_a\mathcal{M}$?

Answer 1

If you mean that $P$ is parallel transport along a fixed curve $\gamma$ then yes. You can apply directly the uniqueness result about parallel transport, i.e. given a curve $\gamma:I \rightarrow M$, $t_0 \in I$ and $u \in T_{\gamma(t_0)}M$, then there exists a unique parallel vector field $X$, such that $X(t_0)=u$.

Answer 2

In general, it is not true. A standard counterexample is by taking $\mathcal{M}$ to be the round sphere $\mathbb{S}^2$ and take $\gamma_i$, $i=1,2,3$, to be the three sides of a triangle on the sphere where each side is an arc of a great circle.

For example, we can take $a,c$ to be two points on the equator and $b$ to be the north pole such that the arc $ac=\gamma_3$ is 1/4 of the equator. Now if we take $v$ to be pointing in the direction of the arc $ab=\gamma_1$, then it can be seen that $P^c_bP^b_av\neq P^c_av$. They differ by a rotation of $90$ degrees.

(Sorry I don't have a photo here, but it is not difficult to have the picture in mind.)

Answer 3

Here is a picture to help you understand