Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-26T05:10:39.669Z Has data issue: false hasContentIssue false
  • English
  • Français

The Classification of 7- and 8-dimensional Naturally Reductive Spaces

Part of: Local differential geometry Global differential geometry

Published online by Cambridge University Press:  30 May 2019

Reinier Storm
Reinier Storm*
Affiliation:
KU Leuven, Department of Mathematics, Celestijnenlaan 200B – Box 2400, BE-3001 Leuven, Belgium Email: reinier.storm@kuleuven.be
Get access Rights & Permissions [Opens in a new window]

Abstract

A new method for classifying naturally reductive spaces is presented. This method relies on a new construction and the structure theory of naturally reductive spaces recently developed by the author. This method is applied to obtain the classification of all naturally reductive spaces in dimension 7 and 8.

Keywords

naturally reductivehomogeneous spaceparallel skew torsionclassification

MSC classification

Primary: 53C30: Homogeneous manifolds
Secondary: 53B20: Local Riemannian geometry
Type
Article
Information
Canadian Journal of Mathematics , Volume 72 , Issue 5 , October 2020 , pp. 1246 - 1274
Check for updates
Copyright
© Canadian Mathematical Society 2019

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

The author is supported by project 3E160361 of the KU Leuven Research Fund.

References

Agricola, I., Ferreira, A. C., and Friedrich, T., The classification of naturally reductive homogeneous spaces in dimensions n⩽6. Differential Geom. Appl. 39(2015), 59–92. https://doi.org/10.1016/j.difgeo.2014.11.005CrossRefGoogle Scholar
Agricola, I., Ferreira, A. C., and Storm, R., Quaternionic Heisenberg groups as naturally reductive homogeneous spaces. Int. J. Geom. Methods Mod. Phys. 12(2015), 1560007, 10. https://doi.org/10.1142/S0219887815600075Google Scholar
Ambrose, W. and Singer, I. M., On homogeneous Riemannian manifolds. Duke Math. J. 25(1958), 647–669.CrossRefGoogle Scholar
Bröcker, T. and tom Dieck, T., Representations of compact Lie groups. Graduate Texts in Mathematics, 98, Springer, New York, 1985. https://doi.org/10.1007/978-3-662-12918-0CrossRefGoogle Scholar
Cartan, E., Sur une classe remarquable d’espaces de Riemann. Bull. Soc. Math. France 54(1926), 214–264. http://www.numdam.org/item?id=BSMF_1926__54__214_0 10.24033/bsmf.1105CrossRefGoogle Scholar
D’Atri, J. E. and Ziller, W., Naturally reductive metrics and Einstein metrics on compact Lie groups. Mem. Amer. Math. Soc. 18(1979), 215, iii + 72.Google Scholar
Friedrich, T., Kath, I., Moroianu, A., and Semmelmann, U., On nearly parallel G 2-structures. J. Geom. Phys. 23(1997), 259–286. https://doi.org/10.1016/S0393-0440(97)80004-6CrossRefGoogle Scholar
Gordon, C. S., Naturally reductive homogeneous Riemannian manifolds. Canad. J. Math. 37(1985), 467–487. https://doi.org/10.4153/CJM-1985-028-2CrossRefGoogle Scholar
Kaplan, A., On the geometry of groups of Heisenberg type. Bull. London Math. Soc. 15(1983), 35–42. https://doi.org/10.1112/blms/15.1.35CrossRefGoogle Scholar
Klaus, S., Diploma Thesis (in German): Einfach-zusammenhängende kompakte homogene Räume bis zur Dimension 9. 06 1988. https://doi.org/10.13140/2.1.4088.8324CrossRefGoogle Scholar
Kostant, B., On differential geometry and homogeneous spaces. I, II. Proc. Natl Acad. Sci. USA 42(1956), 258–261, 354–357.CrossRefGoogle Scholar
Kowalski, O., Counterexample to the “second Singer’s theorem”. Ann. Global Anal. Geom. 8(1990), 2, 211–214. https://doi.org/10.1007/BF00128004CrossRefGoogle Scholar
Kowalski, O. and Vanhecke, L., Four-dimensional naturally reductive homogeneous spaces. Rend. Sem. Mat. Univ. Politec. Torino, (Special Issue): 223–232 (1984), 1983. Conference on differential geometry on homogeneous spaces (Turin, 1983).Google Scholar
Kowalski, O. and Vanhecke, L., Classification of five-dimensional naturally reductive spaces. Math. Proc. Cambridge Philos. Soc. 97(1985), 3, 445–463. https://doi.org/10.1017/S0305004100063027CrossRefGoogle Scholar
Nomizu, K., Invariant affine connections on homogeneous spaces. Amer. J. Math. 76(1954), 33–65.CrossRefGoogle Scholar
Storm, R., A new construction of naturally reductive spaces. Transform. Groups 23(2018), 2, 527–553. https://doi.org/10.1007/s00031-017-9446-5CrossRefGoogle Scholar
Storm, R., Structure theory of naturally reductive spaces. Differential Geom. Appl. 64(2019), 174–200. http://www.sciencedirect.com/science/article/pii/S0926224518302316CrossRefGoogle Scholar
Tricerri, F., Locally homogeneous Riemannian manifolds. Rend. Sem. Mat. Univ. Politec. Torino 50(1993), 4, 411–426. 1992. Differential geometry (Turin, 1992).Google Scholar
Tricerri, F. and Vanhecke, L., Homogeneous structures on Riemannian manifolds. London Mathematical Society Lecture Note Series, 83, Cambridge University Press, Cambridge, 1983. https://doi.org/10.1017/CBO9781107325531CrossRefGoogle Scholar
Tsukada, K., Totally geodesic hypersurfaces of naturally reductive homogeneous spaces. Osaka J. Math. 33(1996), 3, 697–707. http://projecteuclid.org/euclid.ojm/1200787097Google Scholar
Wilking, B., The normal homogeneous space $(\text{SU}(3)\times \text{SO}(3))/\text{U}^{\bullet }(2)$has positive sectional curvature. Proc. Amer. Math. Soc. 127(1999), 4, 1191–1194. https://doi.org/10.1090/S0002-9939-99-04613-4CrossRefGoogle Scholar