Let $c$ be a $W^{1,2}$-curve into a (compact Riemannian) manifold $Q,$ defined on some open interval $I$. Let $t\_0\in I$ and $\xi\_0\in T\_{c(t\_0)}Q$ be arbitrary. I am looking for a citeable reference for the following statement: There is a unique $W^{1,2}$-vector field $\xi$ along $c$ satisfying


(i) $\nabla\_{\dot{c}}\xi=0$ a.e. on $I$


(ii) $\xi(t\_0)=\xi\_0.$


Notice that the statement above could be deduced from the standard Picard-Lindelöf theorem if $c$ was sufficiently regular (i.e. e.g. $c$ was $C^1$).



parallel transport along $W^{1,2}$-curves
Since $W^{1,2} \subset C^0$ and the zero-th order term of (i) depends linearly on $\dot{c} \in L^2$, the usual rewrite-as-integral-equation proof seems to work and is rather straightforward. I don't recall seeing this written down anywhere, but it's easy to verify and summarize.


