Timeline for Is there a geometric interpretation for this quantity?
Current License: CC BY-SA 3.0
14 events
| when toggle format | what | by | license | comment | |
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| Sep 12, 2017 at 6:08 | vote | accept | Ali Taghavi | ||
| Sep 11, 2017 at 23:12 | answer | added | Robert Bryant | timeline score: 12 | |
| Sep 11, 2017 at 20:35 | history | edited | Ali Taghavi | CC BY-SA 3.0 |
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| Sep 11, 2017 at 20:33 | comment | added | Ali Taghavi | @RobertBryant Thank you. I revise it again. | |
| Sep 11, 2017 at 16:39 | comment | added | Robert Bryant | Your question is better now, but why not just say that the sup is to be taken over all $1$-forms $\alpha$ of $g$-length $1$? Thinking of the vector field $X$ doesn't seem to be relevant. | |
| Sep 11, 2017 at 16:28 | history | edited | Ali Taghavi |
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| Sep 11, 2017 at 16:27 | history | undeleted | Ali Taghavi | ||
| Sep 11, 2017 at 14:47 | history | deleted | Ali Taghavi | via Vote | |
| Sep 11, 2017 at 13:59 | comment | added | Ali Taghavi | @RobertBryant Now I revise my question. Could I clarify my question? | |
| Sep 11, 2017 at 13:41 | history | edited | Ali Taghavi | CC BY-SA 3.0 |
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| Sep 11, 2017 at 13:36 | comment | added | Ali Taghavi | I am sorry for not paying attention to the Euler characteristic of odd dimensional manifold. Moreover, to every unit length vector field $X$ we associate a i-form $\alpha_X$ with $\alpha_X(Y)=<X,Y>$. Then we take $\sup$ over all possible unit length vector fields $X$. | |
| Sep 11, 2017 at 13:25 | comment | added | Robert Bryant | First, the Euler characteristic of any compact $3$-manifold without boundary is $0$, so this is no restriction. Second, if $\alpha(Y) = g(X,Y)$ for all vector fields $Y$, then $\alpha = X^\flat$, so there is only one such $\alpha$ for any given unit vector field $X$. Can you reformulate your question? Did you want to just require that $\alpha(X)=1$, which still allows for some variation in $\alpha$? Now the metric is irrelevant. It only depends on $X$. | |
| Sep 11, 2017 at 13:23 | history | edited | Ali Taghavi | CC BY-SA 3.0 |
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| Sep 11, 2017 at 13:15 | history | asked | Ali Taghavi | CC BY-SA 3.0 |