Timeline for Maximally symmetric hyperbolic 3-manifolds with finite volume
Current License: CC BY-SA 4.0
12 events
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| May 21, 2020 at 21:26 | comment | added | HJRW | @AlexArvanitakis: there are many such examples with finite volume. Hyperbolic manifolds are a huge topic of research in low-dimensional topology. The simplest 3-dimensional example with finite volume is perhaps the complement of the figure-eight knot in the 3-sphere (see: en.wikipedia.org/wiki/Figure-eight_knot_(mathematics)). The simplest compact example is perhaps Seifert--Weber dodecahedral space (see en.wikipedia.org/wiki/Seifert%E2%80%93Weber_space). | |
| May 21, 2020 at 20:10 | vote | accept | layman | ||
| May 20, 2020 at 19:39 | comment | added | AlexArvanitakis | Equivalently, such locally maximally symmetric spaces admit the maximum number of Killing vector fields on some neighbourhood of every point. | |
| May 20, 2020 at 19:37 | comment | added | AlexArvanitakis | In GR we sometimes say "locally maximally symmetric" to mean: for every point on the manifold there exists an open neighbourhood of said point which is isometrically diffeomorphic to an open set on a "maximally symmetric" space. So for the purposes of this question said maximally symmetric space would be $H^3$. There should be lots of examples* which are not $H^3$ itself, but whether any have finite volume, I do not know. (*In the Lorentzian case the BTZ black hole geometry is some quotient of maximally symmetric $AdS_3$ but I do not remember the details now.) | |
| May 20, 2020 at 19:27 | answer | added | ThiKu | timeline score: 2 | |
| May 20, 2020 at 17:17 | comment | added | YCor | In any case: finite volume manifolds with curvature $-1$ usually have a very small amount of symmetries, whence the notion of locally symmetric space. | |
| May 20, 2020 at 17:01 | history | edited | YCor |
edited tags
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| May 20, 2020 at 16:58 | comment | added | layman | Sorry, but less popular in standard cosmology, because it is generally believed that space-time in general relativity is not negatively curved. | |
| May 20, 2020 at 16:53 | comment | added | YCor | You probably want to read about Thurston geometries. Also about the notion of "locally symmetric space". That $H^3$ is "less popular"? well, hyperbolic 3-dimensional geometry has been a major theme in geometric topology in the last 40 years! | |
| May 20, 2020 at 16:52 | history | edited | YCor | CC BY-SA 4.0 |
fixed typo, added tag
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| May 20, 2020 at 16:51 | review | First posts | |||
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| May 20, 2020 at 16:50 | history | asked | layman | CC BY-SA 4.0 |