Abstract
We consider invariant Riemannian metrics on compact homogeneous spaces G/H where an intermediate subgroup K between G and H exists. In this case, the homogeneous space G/H is the total space of a Riemannian submersion. The metrics constructed by shrinking the fibers in this way can be interpreted as metrics obtained from a Cheeger deformation and are thus well known to be nonnegatively curved. On the other hand, if the fibers are homothetically enlarged, it depends on the triple of groups (H, K, G) whether non-negative curvature is maintained for small deformations. Building on the work of Schwachhöfer and Tapp (J. Geom. Anal. 19(4):929–943, 2009), we examine all G-invariant fibration metrics on G/H for G a compact simple Lie group of dimension up to 15. An analysis of the low dimensional examples provides insight into the algebraic criteria that yield continuous families of non-negative sectional curvature.
Similar content being viewed by others
References
Brown N., Finck R., Spencer M., Tapp K., Wu Z.: Invariant metrics with nonnegative curvature on compact Lie groups. Canad. Math. Bull. 50(1), 24–34 (2007)
Dynkin, E.B.: Semisimple subalgebras of the semisimple Lie algebras. Mat. Sb. 30, 349–462 (1952) (Russian); English translation: Amer. Math. Soc. Transl. Ser. 2, 6, 111–244 (1957)
Kollross A.: A classification of hyperpolar and cohomogeneity one actions. Trans. Amer. Math. Soc. 354, 571–612 (2002)
Kollross A.: Low cohomogeneity and polar actions on exceptional compact Lie groups. Transform. Groups 14(2), 387–415 (2009)
Schwachhöfer L.: A remark on left invariant metrics on compact Lie groups. Arch. Math. 90, 158–162 (2008)
Schwachhöfer L., Tapp K.: Homogeneous metrics with nonnegative curvature. J. Geom. Anal. 19(4), 929–943 (2009)
Wilking, B.: Nonnegatively and positively curved manifolds. In: Grove, K., Cheeger, J. (eds.) Metric and Comparison Geometry. Surveys in Differential Geometry, vol. 11. International Press, Somerville (2007)
Ziller, W.: Examples of Riemannian manifolds with non-negative sectional curvature. In: Grove, K., Cheeger, J. (eds.) Metric and Comparison Geometry. Surveys in Differential Geometry, vo. 11, International Press, Somerville (2007)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Kerr, M.M., Kollross, A. Nonnegatively curved homogeneous metrics in low dimensions. Ann Glob Anal Geom 43, 273–286 (2013). https://doi.org/10.1007/s10455-012-9345-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10455-012-9345-x