Nearly Kähler and nearly parallel G 2 -structures on spheres

Thomas Friedrich

Archivum Mathematicum (2006)

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

Abstract

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In some other context, the question was raised how many nearly Kähler structures exist on the sphere 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 λ = 12 of the Laplacian acting on 2 -forms. A similar result concerning nearly parallel G 2 -structures on the round sphere 𝕊 7 holds, too. An alternative proof by Riemannian Killing spinors is also indicated.

How to cite

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Friedrich, 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

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  1. Alexandrov B., Friedrich, Th., Schoemann N., Almost hermitian 6 -manifolds revisited, J. Geom. Phys. 53 (2005), 1–30. Zbl1075.53036MR2102047
  2. Brown R. B., Gray A., Vector cross products, Comment. Math. Helv. 42 (1967), 222–236. (1967) Zbl0155.35702MR0222105
  3. Friedrich, Th., Kath I., Moroianu A., Semmelmann U., On nearly parallel G 2 -structures, J. Geom. Phys. 23 (1997), 259–286. (1997) MR1484591
  4. Gray A., Vector cross products on manifolds, Trans. Amer. Math. Soc. 141 (1969), 465–504. (1969) Zbl0182.24603MR0243469
  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. (1969) Zbl0192.59002MR0261515
  6. 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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