Let $X$ and $Y$ be closed Riemannian manifolds and $f,g\colon X\to Y$ two $C^1$-mappings.

Assume that for every $x\in X$ the points $f(x)$ and $g(x)$ can be joined by a unique shortest geodesic of $Y$. Denote this geodesic by $c_x$. Moreover denote by

$P^{c_x}\colon T_{f(x)}Y \to T_{g(x)}Y$

the parallel transport along $c_x$. We get an induced mapping

$P^c\colon \Gamma_{C^1}(f^*TY)\to \Gamma_{C^1}(g^*TY)$

between the $C^1$-sections of the pullback bundles $f^*TY$ and $g^*TY$.

**Question:** Is it true that there exists some constant $C(\|f\|_{C^1},\|g\|_{C^1})>0$ that depends only on the $C^1$-norms of $f$ and $g$ and the geometry of $X$ and $Y$ s.t.

$\|P^c\varphi\|_{\Gamma_{C^1}}\le C(\|f\|_{C^1},\|g\|_{C^1})\|\varphi\|_{\Gamma_{C^1}}$

for all $\varphi\in\Gamma_{C^1}(f^*TY)$? Here, $\|.\|_{\Gamma_{C^1}}$ denotes the usual $C^1$-norm on the respective bundles. Moreover, does the same hold for the first order Sobolev spaces, i.e.

$\|P^c\varphi\|_{\Gamma_{W^1_p}}\le C(\|f\|_{C^1},\|g\|_{C^1})\|\varphi\|_{\Gamma_{W^1_p}}$

for all $\varphi\in\Gamma_{W^1_p}(f^*TY)$?

My intuition says yes, because parallel transport is the solution of an ODE and we have theorems about the differentiable dependence of solutions of ODE's w.r.t. other parameters. The "other parameters" are $f$ and $g$ in this case. However I am not sure if that means that the above inqueality holds since it is a little specific.