Nonnegatively curved homogeneous metrics in low dimensions

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 nonnegative curvature is maintained for small deformations. Building on the work of L. Schwachhöfer and K. Tapp \cite{ST}, 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 nonnegative sectional curvature.

14 pages, to appear in Annals of Global Analysis and Geometry

References in corpus (1)

• Low Cohomogeneity and Polar Actions on Exceptional Compact Lie Groups

Cited by in corpus (2)

• The concept of Cheeger deformations on fiber bundles with compact structure group
• Nonnegatively curved homogeneous metrics obtained by scaling fibers of submersions