User ngrosse - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T06:43:44Z http://mathoverflow.net/feeds/user/13161 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/113781/characterization-of-bounded-geometry-reference-request Characterization of bounded geometry - Reference-request ngrosse 2012-11-18T23:14:20Z 2012-11-19T14:12:52Z <p>I already asked this question at stackexchange three days ago. Since I got no answer, I want to try mathoverflow now. I hope that you can help.</p> <p>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).</p> <p>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.</p> <p>In the paper of Shubin it is written that there are the following equivalent characterizations: (Let $(M,g)$ always have positive injectivity radius.)</p> <p>(i) $(M,g)$ is of bounded geometry.</p> <p>(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 $x′$).</p> <p>I know that (i) is equivalent to the following: (iii) Let $g^\alpha_{ij}$ be the representation of the metric coefficienst on a ball of radius $r$ around $x_\alpha$ with respect to geodesic normal coordinates. Then $g^\alpha_{ij}$ and all its derivatives are uniformly bounded (where uniformly means independent on $\alpha$).</p> <p>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.</p> <p>Thank you in advance.</p> http://mathoverflow.net/questions/113781/characterization-of-bounded-geometry-reference-request/113833#113833 Comment by ngrosse ngrosse 2012-11-19T19:15:07Z 2012-11-19T19:15:07Z Thanks a lot. Do you know any (&quot;citeable&quot;) reference for this? (The book of Eichhorn (mentioned above by Peter Michor) does not work with transition functions and in the PhD-Thesis of Elderling I only found the implication &quot;(i) implies (ii)&quot;)