# Nearly Kähler and nearly parallel ${G}_{2}$-structures on spheres

Archivum Mathematicum (2006)

- Volume: 042, Issue: 5, page 241-243
- ISSN: 0044-8753

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topFriedrich, Thomas. "Nearly Kähler and nearly parallel $G_2$-structures on spheres." Archivum Mathematicum 042.5 (2006): 241-243. <http://eudml.org/doc/249787>.

@article{Friedrich2006,

abstract = {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.},

author = {Friedrich, Thomas},

journal = {Archivum Mathematicum},

keywords = {nearly Kähler structures; nearly parallel $\mathrm \{G\}_2$-structures; nearly Kähler structures; nearly parallel G-structures},

language = {eng},

number = {5},

pages = {241-243},

publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},

title = {Nearly Kähler and nearly parallel $G_2$-structures on spheres},

url = {http://eudml.org/doc/249787},

volume = {042},

year = {2006},

}

TY - JOUR

AU - Friedrich, Thomas

TI - Nearly Kähler and nearly parallel $G_2$-structures on spheres

JO - Archivum Mathematicum

PY - 2006

PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno

VL - 042

IS - 5

SP - 241

EP - 243

AB - 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.

LA - eng

KW - nearly Kähler structures; nearly parallel $\mathrm {G}_2$-structures; nearly Kähler structures; nearly parallel G-structures

UR - http://eudml.org/doc/249787

ER -

## References

top- Alexandrov B., Friedrich, Th., Schoemann N., Almost hermitian $6$-manifolds revisited, J. Geom. Phys. 53 (2005), 1–30. Zbl1075.53036MR2102047
- Brown R. B., Gray A., Vector cross products, Comment. Math. Helv. 42 (1967), 222–236. (1967) Zbl0155.35702MR0222105
- Friedrich, Th., Kath I., Moroianu A., Semmelmann U., On nearly parallel ${\mathrm{G}}_{2}$-structures, J. Geom. Phys. 23 (1997), 259–286. (1997) MR1484591
- Gray A., Vector cross products on manifolds, Trans. Amer. Math. Soc. 141 (1969), 465–504. (1969) Zbl0182.24603MR0243469
- 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. (1969) Zbl0192.59002MR0261515
- Grunewald R., Six-dimensional Riemannian manifolds with a real Killing spinor, Ann. Global Anal. Geom. 8 (1990), 43–59. (1990) Zbl0704.53050MR1075238

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