Parallel transport respects concatenation of paths

Let $E\to M$ be a vector bundle equipped with a connection on a vector bundle $\nabla$. For a piecewise smooth path $\gamma:[0,1]\to M$, write

for the parallel transport map determined by $\nabla$.

Proposition (compatibility with concatenation)

Let $\gamma_1:[0,1]\to M$ and $\gamma_2:[0,1]\to M$ be piecewise smooth with $\gamma_1(1)=\gamma_2(0)$. Let $\gamma_2\ast \gamma_1$ denote their concatenation (first traverse $\gamma_1$, then $\gamma_2$). Then

Equivalently, if $s(t)$ is a $\nabla$-parallel section along $\gamma_2\ast\gamma_1$ with initial value $s(0)=v\in E_{\gamma_1(0)}$, then the value at the intermediate point is $s(1/2)=\mathrm{PT}_{\gamma_1}(v)$ (after reparametrization), and the final value is obtained by transporting further along $\gamma_2$.

The same statement holds for parallel transport in a principal $G$-bundle with a principal connection: transport along a concatenated path is the composition of the two transport maps between fibers.

Examples

1. Trivial connection on $M\times V$. If $\nabla=d$, then $\mathrm{PT}_\gamma$ is the identity on $V$ for every $\gamma$, so the concatenation identity holds tautologically.
2. Piecewise geodesic transport. On a Riemannian manifold with its Levi–Civita connection, transporting a tangent vector along a broken geodesic is the same as transporting along the first segment and then along the second; the proposition encodes this “do it in steps” rule.
3. Holonomy as a product. For a loop written as a concatenation of subloops, the resulting holonomy element is the product of the holonomies of the pieces (with the same order as the concatenation).