Timeline for Bounding Riemannian Distance
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 12, 2017 at 19:17 | comment | added | ABIM | Oh you're right. I was thinking something similar today but how can I use comparison triangels to prove (the reverse) type of inequality? | |
| Sep 11, 2017 at 9:44 | comment | added | Sebastian Goette | Sorry, my comment above is rubbish. Rather, note two things. First, if $M$ has $\pi_1(M)\ne\{0\}$, then it is always possible that $d_M(x,y)$ is very small, even though preimages of $x$ and $y$ in $T_{x_0}M$ can be far apart. Second, the inequality from the very left (with a square inserted) to the very right is wrong. And it should be clear by considering very small triangles that as stated (without the square), the first inequality cannot hold. | |
| Sep 10, 2017 at 17:09 | comment | added | Sebastian Goette | For the first inequality, you want to map the "comparison triangle" in $(TM,g_H)$ to $M$. This gives you an inequality similar *but not identical!) to your first. You get an inequality similar to the second by regarding cosine theorems for hyperbolic and Euclidean geometry - maybe you can figure this out for yourself? | |
| Sep 9, 2017 at 17:28 | history | edited | ABIM | CC BY-SA 3.0 |
added 10 characters in body
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| Sep 9, 2017 at 8:08 | history | edited | Ben McKay | CC BY-SA 3.0 |
spelling, grammar
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| Sep 9, 2017 at 8:02 | comment | added | Ben McKay | I thnk you want to square the first $d_H$, so that this might be true when $M$ is hyperbolic space. Also, you want ${}\le{}$ instead of ${}<{}$. | |
| Sep 9, 2017 at 4:08 | comment | added | Anton Petrunin | The inequality has no sense --- please red/correct it. | |
| Sep 9, 2017 at 0:18 | history | asked | ABIM | CC BY-SA 3.0 |