\mathsf S¹-actions on 4-manifolds and fixed point homogeneous manifolds of nonnegative curvature

2015/10/06 by Spindeler, Wolfgang
#53 #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1510.01548

Abstract

This is a slightly altered version of the authors thesis from 2014. In the first main part we show that the quotient space of a compact, simply connected and nonnegatively curved Riemannian 4-manifold by an effective, isometric circle-action admits an approximation in Gromov-Hausdorff topology by smooth, positively curved Riemannian 3-manifolds. In the second main part we show that a compact, nonnegatively curved and fixed point homogeneous manifold is diffeomorphic to the unit normal bundles of two smooth closed submanifolds glued together along their boundaries. As a corollary we show that a compact and simply connected torus manifold admitting an invariant metric of nonnegative curvature is rationally elliptic.

Related

• Nonnegatively curved fixed point homogeneous manifolds in low dimensions
• On positively curved 4-manifolds with S¹-symmetry
• Simply connected Alexandrov 4-manifolds with positive or nonnegative curvature and torus actions
• Curvature homogeneous manifolds in dimension 4
• Torus Actions, Fixed-Point Formulas, Elliptic Genera and positive curvature