Abstract
We introduce a geometric invariant that we call the index of symmetry, which measures how far is a Riemannian manifold from being a symmetric space. We compute, in a geometric way, the index of symmetry of compact naturally reductive spaces. In this case, the so-called leaf of symmetry turns out to be of the group type. We also study several examples where the leaf of symmetry is not of the group type. Interesting examples arise from the unit tangent bundle of the sphere of curvature 2, and two metrics in an Aloff-Wallach 7-manifold and the Wallach 24-manifold.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Alekseevskii, D.V. (originator): Reductive space. Encyclopedia of Mathematics, http://www.encyclopediaofmath.org/index.php?title=Reductive_space&oldid=11232
Aloff, S., Wallach, N.: An infinite family of distinct 7-manifolds admitting positively curved Riemannian structures. Bull. Am. Math. Soc. 81, 93–95 (1975)
Berndt, J., Console, S., Olmos, C.: Submanifolds and holonomy. In: Research Notes in Mathematics, vol. 434. Chapman & Hall/CRC, Boca Raton (2003)
Cartan, É.: Sur une classe remarquable d’spaces de Riemann. I. Bull. Soc. Math. France 54, 214–264 (1926)
Cartan, É.: Sur une classe remarquable d’spaces de Riemann. II. Bull. Soc. Math. France 55, 114–134 (1927)
Eschenburg, J.-H., Olmos, C.: Rank and symmetry of Riemannian manifolds. Comment. Math. Helv. 69(3), 483–499 (1994)
Kostant, B.: On differential geometry and homogeneous spaces. II. Proc. Natl. Acad. Sci. USA 42, 354–357 (1956)
Olmos, C., Reggiani, S.: A note on the uniqueness of the canonical connection of a naturally reductive space. Monatsh. Math. 172, 379–386 (2013)
Olmos, C., Reggiani, S.: The skew-torsion holonomy theorem and naturally reductive spaces. J. Reine Angew. Math. 664, 29–53 (2012)
Reggiani, S.: On the affine group of a normal homogeneous manifold. Ann. Glob. Anal. Geom. 37(4), 351–359 (2010)
Reggiani, S.: A Berger-type theorem for metric connections with skew-symmetric torsion. J. Geom. Phys. 65, 26–34 (2013)
Acknowledgments
The work of C. Olmos and S. Reggiani was supported by Universidad Nacional de Córdoba and CONICET, and partially supported by ANCyT, Secyt-UNC and CIEM. H. Tamaru was supported in part by KAKENHI (24654012).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Olmos, C., Reggiani, S. & Tamaru, H. The index of symmetry of compact naturally reductive spaces. Math. Z. 277, 611–628 (2014). https://doi.org/10.1007/s00209-013-1268-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00209-013-1268-0