| Title:
|
Nearly Kähler and nearly parallel $G\sb 2$-structures on spheres (English) |
| Author:
|
Friedrich, Thomas |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
42 |
| Issue:
|
5 |
| Year:
|
2006 |
| Pages:
|
241-243 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
In some other context, the question was raised how many nearly Kähler structures exist on the sphere $\mathbb {S}^6$ equipped with the standard Riemannian metric. In this short note, we prove that, up to isometry, there exists only one. This is a consequence of the description of the eigenspace to the eigenvalue $\lambda = 12$ of the Laplacian acting on $2$-forms. A similar result concerning nearly parallel $\mathrm {G}_2$-structures on the round sphere $\mathbb {S}^7$ holds, too. An alternative proof by Riemannian Killing spinors is also indicated. (English) |
| Keyword:
|
nearly Kähler structures |
| Keyword:
|
nearly parallel $\mathrm {G}_2$-structures |
| MSC:
|
53C15 |
| MSC:
|
53C29 |
| idZBL:
|
Zbl 1164.53353 |
| idMR:
|
MR2322410 |
| . |
| Date available:
|
2008-06-06T22:49:34Z |
| Last updated:
|
2012-05-10 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/108030 |
| . |
| Reference:
|
[1] Alexandrov B., Friedrich, Th., Schoemann N.: Almost hermitian $6$-manifolds revisited.J. Geom. Phys. 53 (2005), 1–30. Zbl 1075.53036, MR 2102047 |
| Reference:
|
[2] Brown R. B., Gray A.: Vector cross products.Comment. Math. Helv. 42 (1967), 222–236. Zbl 0155.35702, MR 0222105 |
| Reference:
|
[3] Friedrich, Th., Kath I., Moroianu A., Semmelmann U.: On nearly parallel $\mathrm{G}_2$-structures.J. Geom. Phys. 23 (1997), 259–286. MR 1484591 |
| Reference:
|
[4] Gray A.: Vector cross products on manifolds.Trans. Amer. Math. Soc. 141 (1969), 465–504. Zbl 0182.24603, MR 0243469 |
| Reference:
|
[5] Gray A.: Six-dimensional almost complex manifolds defined by means of three-fold vector cross products.Tohoku Math. J. II. Ser. 21 (1969), 614–620. Zbl 0192.59002, MR 0261515 |
| Reference:
|
[6] Grunewald R.: Six-dimensional Riemannian manifolds with a real Killing spinor.Ann. Global Anal. Geom. 8 (1990), 43–59. Zbl 0704.53050, MR 1075238 |
| . |
