How to stream radio from the command line?
- Express that you have enjoyed working in your department. (Even if this is not entirely true.)
- Explain that you are only leaving because of an opportunity "too exciting to turn down." Don't volunteer exactly why it is exciting; if asked, talk about what you'll be doing, not what you'll be paid.
- If there's a way you can help the department while you're in industry -- serving on a committee, helping with accreditation, accepting an intern, things like that -- it would be kind of you (though it is not obligatory) to volunteer unasked.
- Keep your resignation letter short and cordial.
These tactics should suffice to keep your departure as amicable as it can be, assuming you're doing the normal things to minimize disruption (e.g. not leaving mid-term).
There is no such thing as "dofollow". All links are followed unless specifically stated otherwise (nofollow).
As Deane Yang points out, you want bounds from below on the injectivity radius and from above on the curvature. You may want to consider a manifold of bounded geometry, which means that in addition all derivatives of the curvature tensor are bounded. This last condition is equivalent to having an atlas of coordinate charts such that all transition functions have uniformly bounded derivatives.
This topic is discussed in a very down-to-earth way in Appendix 1 of the paper "Spectral Theory Of Elliptic Operators On Non-Compact Manifolds" by Shubin. Your question is addressed in Lemma 1.2 on page 30 and Lemma 1.3 on page 31. Here is a link to the paper:
http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.27.4473