Timeline for Deformed completion of negatively curved metric
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
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| May 28 at 18:46 | comment | added | Igor Belegradek | @Yasha: I don't think this kind of construction is known. The only cases I know is when you remove a totally geodesic compact surface from a locally symmetric negatively curved 4-manifold, see arxiv.org/abs/1609.01974, arxiv.org/abs/0711.5001, arxiv.org/abs/0711.2324. The case when you remove a compact totally geodesic surface from an arbitrary negative curved $4$-manifold is open (to my knowledge). | |
| May 28 at 18:16 | comment | added | Yasha | @Igor Belegradek Can one use this kind of construction to show that $(\mathbb{R} ^{2} - \{0\}) \times \Sigma$, for $\Sigma $ a compact negatively curved Riemann surface has negative sectional curvature metric? | |
| May 27 at 14:37 | history | edited | Yasha | CC BY-SA 4.0 |
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| May 25 at 19:23 | comment | added | Deane Yang | Please edit your question so that it doesmake sense. | |
| May 25 at 16:33 | history | edited | Yasha | CC BY-SA 4.0 |
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| May 25 at 13:56 | comment | added | Moishe Kohan | I was only thinking about totally geodesic submanifolds. | |
| May 25 at 13:53 | comment | added | Igor Belegradek | @MoisheKohan: the OP makes no assumption about $C$. Say, what if $C$ is a codimension 2 sphere? If the curvature is constant near $C$, and $C$ is totally geodesic of codimension 2, the deformation exists and is quite straightforward. There are similar (less straightforward) results when $X$ is locally symmetric and $C$ is totally geodesic of codimension 2; in this case the normal bundle of $C$ won't be trivial, but this is irrelevant. For the most difficult case see arxiv.org/abs/1609.01974 which has references to easier cases. | |
| May 25 at 13:44 | comment | added | Moishe Kohan | @IgorBelegradek: apart from the constant curvature case. | |
| May 25 at 13:29 | comment | added | Igor Belegradek | Even if $C$ has codimension 2, I am not aware of any such deformation. | |
| May 25 at 13:19 | comment | added | Moishe Kohan | You should assume that $C$ has codimension 2. | |
| May 25 at 11:52 | comment | added | Igor Belegradek | If $X$ is the hyperbolic $n$-space with $n>2$ and $C$ is a point, then then $X-C$ is not aspherical, so has no complete negatively curved metric (by Cartan-Hadamard). | |
| May 25 at 11:28 | history | asked | Yasha | CC BY-SA 4.0 |