# Basic results in bounded geometry

I'm doing analysis (dynamical systems) in the context of Riemannian manifolds of bounded geometry and I find myself reproving quite a few standard results/tools from standard differential geometry, such as locally finite covers and subordinate partitions of unity, a tubular neighborhood theorem, smoothing of submanifolds...

The main difference from the standard results is that I require uniformly bounded estimates, so for example the tubular neighborhood must have a uniformly finite size and a uniformly bounded diffeomorphism. This means that I cannot simply generalize the standard proofs.

I'm not familiar with bounded geometry. The only reference with explicit details I found is Schick: Manifolds with Boundary and of Bounded Geometry where a uniformly locally finite cover with subordinate partition of unity is proven.

Are there other books or articles which include similar results in the context of bounded geometry?

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I don't have any specific references for partitions of unity or tubular neighborhoods, but I can suggest two authors who have written a lot about bounded geometry manifolds and metric spaces: John Roe and Guoliang Yu. You might try "Lectures on Coarse Geometry" by Roe, for example; even if your specific questions are not answered there, I bet it would be fruitful to chase down the references. – Paul Siegel Jun 1 '11 at 14:35
I recommend the books of Jost and Gallot-Hulin-Lafontaine, as well as Gromov's classic "Metric Structures for Riemannian and Non-Riemannian Spaces". For tubular neighborhood, see the classic paper of Heintze-Karcher. For locally finite covers and partitions of unity, check the papers of Stefan Peters and Greene-Wu on convergence of Riemannian manifolds and the references cited by them. – Deane Yang Jun 6 '11 at 14:20

Many results, in particular about Sobolev spaces, for Riemannian manifolds with bounded geometry, are in:

• J. Eichhorn. Global Analysis on Open Manifolds. Nova Science Publishers Inc., New York, 2007.

• H. Triebel. Theory of Function Spaces. II, Volume 84 of Monographs in Mathematics. Birkhauser Verlag, Basel, 1992.

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Thanks, I finally got a hold of the book by Eichhorn, and there are indeed many results and references in there! I haven't found the uniform tubular neighborhood and smoothed manifold theorems yet, so I guess these might be new then. – Jaap Eldering Jun 7 '13 at 14:19