Timeline for Algebraic atlas on smooth manifolds
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 5, 2021 at 10:57 | history | edited | Robert Bryant | CC BY-SA 4.0 |
Edited the last paragraph (converting into two) to make it more clear.
|
| Dec 5, 2021 at 10:39 | comment | added | Robert Bryant | @Zerox: Any connected compact real smooth 2-manifold has a metric of constant curvature $K$, so it has an atlas whose transition maps are isometries between contractible open sets in the simply-connected surface of constant curvature $K$. Since these latter spaces can be presented in such a way that their isometries are rational maps, every compact real smooth 2-manifold has at least one rational structure. I'm not aware of any classification up to rational equivalence of such real rational structures on 2-manifolds, though. The case of geometrizable 3-manifolds is probably similar. | |
| Dec 4, 2021 at 14:01 | comment | added | Ben McKay | Every primary Kodaira surface admits infinitely many holomorphic rational structures not derived from any projective connection: McKay, Benjamin Exotic geometric structures on Kodaira surfaces. Indiana Univ. Math. J. 62 (2013), no. 2, 643–670. | |
| Dec 4, 2021 at 13:53 | comment | added | Zerox | Are there results about the well-undertood case of real $2$-manifolds and geometrizable real $3$-manifolds? | |
| Dec 4, 2021 at 13:48 | vote | accept | Zerox | ||
| Dec 4, 2021 at 13:32 | comment | added | Ben McKay | For the classification of holomorphic projective connections on complex surfaces, see Bruno Klingler, Structures affines et projectives sur les surfaces complexes, Ann. Inst. Fourier (Grenoble) 48 (1998), no. 2, 441–477. MR MR1625606 (99c:32038). But these are not the same as rational structures, since the birational automorphisms of the complex affine plane are more complicated. | |
| Dec 4, 2021 at 13:16 | history | answered | Robert Bryant | CC BY-SA 4.0 |