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.

  • $\begingroup$ Have you tried to work this out using the ODE theorem? As you say, it is a specific case. $\endgroup$ – Deane Yang Jan 10 '17 at 16:38

Counterexample: Let $X=S^1$ and $Y$ defined below. The norm of $P^{c_{t+\epsilon}} - P^{c_{t}}$ is given, essentially, by (Ambrose-Singer) the norm of the parallel transport minus identity operator along the closed curve $s_{t,\epsilon}$ formed by $(f(\tau), [t\leq \tau \leq {t+\epsilon}]), {c_{t+\epsilon}}, (g(\tau), [t\leq \tau \leq {t+\epsilon}]), c_t$; which in term is estimated above by the area of some surface $\Sigma_{t,\epsilon}$ with boundary $ s_{t,\epsilon }$ times curvature of $Y$. The most natural choice is $\Sigma_{t,\epsilon} = \cup c_\tau, ), [t\leq \tau \leq {t+\epsilon}]$. But then, assuming that $\| \dot f(t)\| = \| \dot g(t)\| = 1 $ the derivative of this area on $\epsilon$ is estimated as the ratio $R$ between the maximal and minimal norms of the Jacobi field generated by variation of geodesics $c_\tau$. Now define $Y$ such that the points $f(t)$ and $g(t)$ are conjugated, but the curve $g(\tau)$ intersect cut locuses of $f(\tau)$ only at $g(t)$, and the same is true for the curve $f(\tau)$ (I remember some explicit example by Gromoll). Then all $c_\tau$ are correctly defined and $R(\tau)$ goes to $\infty$ as $\tau$ goes to $t$, and so the $C^1$-norm of $ P^{c_{\tau}}$.

NB. Of course, for $Y$ without conjugate points (say, of nonpositive sectional curvature) your estimate is true - by the same arguments above.


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