Timeline for Smoothness of distance function in Riemannian Manifolds
Current License: CC BY-SA 4.0
13 events
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| Feb 24, 2023 at 21:54 | comment | added | Chee | @ Jan Bohr: never mind. I see why that claim holds: if $z_0 \in U \cap A$, then there is a nonempty, open neighborhood $S$ of $z_0$ such that $ S \subseteq U$ since $U$ is open. Since $z_0 \in A$ and $B$ is dense in $A$, we must have $S \cap B \ne \varnothing$. So, $ \varnothing \ne U \cap A \supseteq S \cap B$ | |
| Feb 24, 2023 at 19:36 | comment | added | Chee | @ Jan Bohr: let $A = \exp_x{(\partial \Sigma_x)} $ and $B = \exp_x{(\partial^1 \Sigma_x)} $. In your Lemma 1, can you please explain why "$U \cap A \ne \varnothing$ implies $U \cap B \ne \varnothing$ when $B$ is dense in $A$"? This claim is the key to prove Lemma 1. We know that $A$ has no interior points with respect to the metric topology on $M$. But what if $U \cap A = \{y_0\}$ for some $y_0 \notin B$, which then implies that $U \cap B = \varnothing)$? Thank you | |
| Apr 26, 2020 at 11:01 | history | edited | Jan Bohr | CC BY-SA 4.0 |
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| Apr 25, 2020 at 17:58 | comment | added | Chee | your (2) is equivalent to Part 2 of Proposition 4.8 from Sakai, except that my emphasis at that time was on "continuation" (which is a widely used technique in analysis of geodesics (including the proof of the Hopf-Rinow theorem) and in complex analysis). wrt benefits to students, i personallly like "continuation" more | |
| Apr 25, 2020 at 17:51 | comment | added | Chee | hi Jan, in the 1st line of your proof of Lemma 1, $d(x,y)$ should be $d(x,y)^2$. i edited it but my edit was rejected. | |
| Apr 25, 2020 at 17:50 | comment | added | Chee | hi Jan, glad that you think my comment is useful. i added a note to your answer by pointing out Klingenberg's Theorem 2.1.12 and some details. but one moderator thought my note makes no sense and rejected my edits to your answer. i like you answer since it is very transparent and instructional (since I am an educator at the same time). on the other hand, denseness cannot be removed and your proof is already good enough. finally, i realized that you probably used a different version of Klingenberg. | |
| Apr 25, 2020 at 17:42 | history | edited | Jan Bohr | CC BY-SA 4.0 |
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| Apr 25, 2020 at 17:41 | comment | added | Jan Bohr | Thank you very much! In fact, Klingenberg's Theorem 2.1.12 does the job, as I explain here. Part 2 from Proposition 4.8 (Sakai) seems to go for the same result as me in equation (2), but his proof seems to be a bit more involved. I don't see that what should be missing with my proof though. | |
| Apr 24, 2020 at 18:28 | comment | added | Chee | wrt to the use of denseness claim in your proof. if we combine: Part 2 of Proposition 4.8 (page 108 of Sakai) and its proof, Theorem 2.1.12 and the first claim of Theorem 2.1.14 (page 133 of Klingenberg) and its proof, we do not need the denseness claim explicitly. however, Theorem 2.1.12 is almost equivalent to the denseness claim. | |
| Apr 24, 2020 at 18:16 | review | Suggested edits | |||
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| Feb 3, 2020 at 10:29 | history | edited | Jan Bohr | CC BY-SA 4.0 |
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| May 23, 2019 at 9:46 | history | edited | Jan Bohr | CC BY-SA 4.0 |
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| May 22, 2019 at 16:31 | history | answered | Jan Bohr | CC BY-SA 4.0 |