# parallel transport along $W^{1,2}$-curves

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 $C^1$).

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$W^{1,2}$ means one derivative in $L_2$? – Deane Yang Oct 19 '10 at 20:28
@Deane: That's correct! – Orbicular Oct 19 '10 at 20:29
And you know how to prove it but would like a reference? – Deane Yang Oct 19 '10 at 20:33
I know a 40 year old refence in German. There the authors claim the statement to be well-known (and still give a proof in the appendix). Hence I thought there might be an English reference as well (since I get the feeling - while reading the paper - that the statement is not original to the paper) – Orbicular Oct 19 '10 at 20:41
Since $W^{1,2} \subset C^0$ and (i) is a linear first ODE, the usual rewrite-as-integral-equation proof seems to work and is rather straightforward. I don't recall seeing this written down anywhere, but, if I'm correct on this, it's easy to verify and summarize. – Deane Yang Oct 19 '10 at 20:49

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.
You will find a proof in Appendix D (Theorem D.1) of my book `A Tour of SubRiemannian Geometry'. It may not look like what you want at first glance, since that theorem is stated in a more general context, applicable to parallel transport in principal bundles. Take the vector fields $X_a$ there to be the standard framing of the horizontal space associated to the Levi-Civita connection, as viewed as a a distribution on the orthonormal frame bundle of your $Q$.