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I'm looking for a proof of an equivalence that can e.g. be found in a paper by Shubin 'Spectral theory of elliptic operators on non-compact manifolds' (Appendix A.1.1 below Def. 1.1).

It's about manifolds of bounded geometry, where bounded geometry means here: positive injectivity radius and the curvature tensor and all its covariant derivatives are bounded.

In the paper of Shubin it is written that there are the following equivalent characterizations: (Let $(M,g)$ always have positive injectivity radius.)

(i) $(M,g)$ is of bounded geometry.

(ii) Let $r>0$ be smaller than the injectivity radius. For any $x,x'\in M$ let $y$ (resp. $y'$) be geodesic normal coordinates around $x$ (resp. $x'$) with an $r$-ball as domain. If the balls of radius $r$ around $x$ and $x'$ have a nonempty intersection, then the transition function $y^{-1}\circ y$ and all its derivatives are uniformly bounded (where uniformly means here independent of $x$ and $x'$.

I know that (i) is equivalent to the following: (iii) Let $g_{ij}^\alpha$ be the representation of the metric coefficienst on a ball of radius $r$ around $x_\alpha$ with respect to geodesic normal coordinates. Then $g_{ij}^\alpha$ and all its derivatives are unifomrly bounded (where uniformly means independent on $\alpha$).

I can see that (iii) implies (ii) (by looking at the geodesic flow and using Gronwall's inequality). But I have no idea how to get the converse. Is there any reference for the proof? I'm grateful for any idea for the proof as well.

Thank you in advance.

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