Geodesic
Nearly Kähler and nearly parallel $G\sb 2$-structures on spheres
Archivum mathematicum, Tome 42 (2006) no. 5, pp. 241-243 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

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.
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.
Classification : 53C15, 53C29
Keywords: nearly Kähler structures; nearly parallel $\mathrm {G}_2$-structures 
@article{ARM_2006_42_5_a10,
     author = {Friedrich, Thomas},
     title = {Nearly {K\"ahler} and nearly parallel $G\sb 2$-structures on spheres},
     journal = {Archivum mathematicum},
     pages = {241--243},
     year = {2006},
     volume = {42},
     number = {5},
     mrnumber = {2322410},
     zbl = {1164.53353},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ARM_2006_42_5_a10/}
}
TY  - JOUR
AU  - Friedrich, Thomas
TI  - Nearly Kähler and nearly parallel $G\sb 2$-structures on spheres
JO  - Archivum mathematicum
PY  - 2006
SP  - 241
EP  - 243
VL  - 42
IS  - 5
UR  - http://geodesic.mathdoc.fr/item/ARM_2006_42_5_a10/
LA  - en
ID  - ARM_2006_42_5_a10
ER  - 
%0 Journal Article
%A Friedrich, Thomas
%T Nearly Kähler and nearly parallel $G\sb 2$-structures on spheres
%J Archivum mathematicum
%D 2006
%P 241-243
%V 42
%N 5
%U http://geodesic.mathdoc.fr/item/ARM_2006_42_5_a10/
%G en
%F ARM_2006_42_5_a10
Friedrich, Thomas. Nearly Kähler and nearly parallel $G\sb 2$-structures on spheres. Archivum mathematicum, Tome 42 (2006) no. 5, pp. 241-243. http://geodesic.mathdoc.fr/item/ARM_2006_42_5_a10/

[1] Alexandrov B., Friedrich, Th., Schoemann N.: Almost hermitian $6$-manifolds revisited. J. Geom. Phys. 53 (2005), 1–30. | MR | Zbl

[2] Brown R. B., Gray A.: Vector cross products. Comment. Math. Helv. 42 (1967), 222–236. | MR | Zbl

[3] Friedrich, Th., Kath I., Moroianu A., Semmelmann U.: On nearly parallel $\mathrm{G}_2$-structures. J. Geom. Phys. 23 (1997), 259–286. | MR

[4] Gray A.: Vector cross products on manifolds. Trans. Amer. Math. Soc. 141 (1969), 465–504. | MR | Zbl

[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. | MR | Zbl

[6] Grunewald R.: Six-dimensional Riemannian manifolds with a real Killing spinor. Ann. Global Anal. Geom. 8 (1990), 43–59. | MR | Zbl