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Suppose $M$ is a manifold with a connection $\nabla$. Fix $p\in M, v\in T_p M$ and let $B$ be a small neighborhood of $p$ such that every $x\in B$ can be joigned with $p$ by a unique geodesic $\gamma_{p,x}$. Each geodesic $\gamma=\gamma_{p,x}$ defines with $v$ a parallel transport $(V_t)_{t\in [0,1]}$ along $\gamma$ such that $V_0=v$. I want to prove that for all $f$ smooth on $M$ $$\sup_{x\in B}\sup_{t\in [0,1]} |V_t f|<\infty$$

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  • $\begingroup$ Since geodesics and parallel transport are defined in coordinates by ODEs, the fact that the connection coefficients are smooth means that you will have continuous dependence of $V_t$ on $x$ and $t$, so you can just throw (pre)compactness at it. $\endgroup$ – Anthony Carapetis Jun 25 '16 at 5:01
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Here is a possible idea. The equation defining $V$ is linear with coefficients depending on $\gamma$ and Christoffel symbols $\Gamma_{i,j}$ and so $V$ is explicitly given by $$V_t=v \exp(-\int_{0}^{t}...)$$ where $...$ is a function of $\gamma'_i$ and $\Gamma_{i,j}$ (we take a local chart). So we only need to prove that for all $i$, $$\sup_{x\in B}\int_{0}^{1} |\gamma_i'(s)| ds<\infty$$ How can we prove this ?

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