Timeline for how to define the injectivity radius of manifolds with boundary?
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
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| Apr 18, 2016 at 8:54 | vote | accept | quarague | ||
| Apr 15, 2016 at 9:42 | comment | added | Sebastian Goette | @ThomasRot This would not work well for manifolds where the boundary is not totally geodesic. For example, if you take the double of the flat unit disk, then two points on the boundary would have two minimizing geodesics between them, one on each part. Neither would the exponential map at a boundary point be injective. So with that definition, you get $0$ again. If you just ask for a unique minimizer, there is no trouble. Schick's definition seems to work as well, but gives $1$ (coming from injectivity of the collar) unless I misinterpreted the statement. | |
| Apr 15, 2016 at 7:26 | history | edited | quarague | CC BY-SA 3.0 |
summmarizing the ideas so far
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| S Apr 14, 2016 at 13:04 | history | suggested | Amir Sagiv |
Added the definitions tag
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| Apr 14, 2016 at 12:42 | review | Suggested edits | |||
| S Apr 14, 2016 at 13:04 | |||||
| Apr 14, 2016 at 10:17 | comment | added | Anton Petrunin | The paper "Geometric curvature bounds in Riemannian manifolds with boundary" by Alexander, Berg and Bishop is related. In particular, it follows a lower bound on your unique-length-minimizer-injectivity radius from upper curvature bound and second fundamental form of the boundary. ams.org/journals/tran/1993-339-02/S0002-9947-1993-1113693-1/… | |
| Apr 14, 2016 at 9:12 | answer | added | user44172 | timeline score: 8 | |
| Apr 14, 2016 at 7:22 | comment | added | Thomas Rot | Maybe one should define it as the injectivity radius of the double of the manifold? | |
| Apr 14, 2016 at 6:48 | history | asked | quarague | CC BY-SA 3.0 |