Nonnegatively curved homogeneous metrics obtained by scaling fibers of submersions

TL;DR

This paper investigates how scaling fibers in Riemannian submersions affects nonnegative curvature on homogeneous spaces, providing classifications for specific group configurations.

Contribution

It classifies group triples where fiber scaling preserves nonnegative curvature, extending understanding of metric deformations on homogeneous spaces.

Findings

Classified triples (H,K,G) maintaining nonnegative curvature under fiber scaling.

Provided a complete classification for H of full rank.

Achieved an almost complete classification for regular subgroups.

Abstract

We consider invariant Riemannian metrics on compact homogeneous spaces G/H where an intermediate subgroup K between G and H exists, so that the homogeneous space G/H is the total space of a Riemannian submersion. We study the question as to whether enlarging the fibers of the submersion by a constant scaling factor retains the nonnegative curvature in the case that the deformation starts at a normal homogeneous metric. We classify triples of groups (H,K,G) where nonnegative curvature is maintained for small deformations, using a criterion proved by Schwachh\"ofer and Tapp. We obtain a complete classification in case the subgroup H has full rank and an almost complete classification in the case of regular subgroups.