Timeline for Extending metrics from $M =\mathbb{T}^2 \times (-\pi , \pi)$ to $ \mathbb{T}^3$
Current License: CC BY-SA 4.0
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 24, 2018 at 14:09 | comment | added | Junior Soares | Can you explain me better why $\theta$ has to be necessarily periodic? | |
| May 17, 2018 at 1:03 | comment | added | Tim Carson | When you said $\theta: (-\pi, \pi)^3 \to \mathbb{R}_+$ you seemed to be implicitly identifying $\mathbb{T}$ with $[-\pi, \pi]/(-\pi \sim \pi)$. With that identification we can say $(-\pi, \pi)^3 \subset \mathbb{T}^3$. | |
| May 15, 2018 at 12:04 | comment | added | Junior Soares | To me there is some issues like that: How may I extends $\theta$ to $\mathbb{T}^3$ if $(\pi, \pi)^3 \not \subset \mathbb{T}^3$? | |
| May 15, 2018 at 12:00 | comment | added | Junior Soares | Let me know if I understand. Are you saying that would be much more aproprieted if it were so: Let $M = \mathbb{T}^2 \times (-\pi , \pi)$ be a manifold with the metric: $$g = r^2 dx^2 + (R + r.cos(x))^2 dy^2 + \theta(x,y,z)^2dz^2$$ and $\theta: (-\pi, \pi)^3 \to \mathbb{R_+}$ so $g$ extends to a metric in $\mathbb{T}^3$ if and only if $g$ extends to a periodic function on $\mathbb{T}^3$ such that $$ \int_{-\pi}^{\pi}\theta(x,y,z)dz < \infty$$ where $A = (-\pi, \pi)^3 $? | |
| May 7, 2018 at 22:33 | history | answered | Tim Carson | CC BY-SA 4.0 |