Nonnegatively curved fixed point homogeneous manifolds in low dimensions

Abstract

Let G be a compact Lie group acting isometrically on a compact Riemannian manifold M with nonempty fixed point set M G . We say that M is fixed-point homogeneous if G acts transitively on a normal sphere to some component of M G . Fixed-point homogeneous manifolds with positive sectional curvature have been completely classified. We classify nonnegatively curved fixed-point homogeneous Riemannian manifolds in dimensions 3 and 4 and determine which nonnegatively curved simply-connected 4-manifolds admit a smooth fixed-point homogeneous circle action with a given orbit space structure.

Nonnegatively curved fixed point homogeneous 5-manifolds

“…Nevertheless, in dimensions at most 5, where fixed point set components have dimension at most 3, a complete topological classification can be given in the simply connected case. In dimensions 4 and below the topological classification was carried out in [6]. In this paper we address the topological classification in dimension 5.…”

Odd-dimensional GKM-manifolds of non-negative curvature

“…We recall the classification of closed, non-negatively curved T 1 -fixed point homogeneous manifolds due to Galaz-García [12]. Theorem 2.38.…”

Nonnegatively curved 5–manifolds with almost maximal symmetry rank

“…By definition, the action is fixed point homogeneous. Closed fixed point homogeneous manifolds 3-manifold with nonnegative curvature were classified in [13] and we recall their classification in the orientable case. Observe first that the fixed point set is 1-dimensional, with at most two components, and these components are circles.…”

“…[5]) that there exists a flat, totally geodesic triangular surface △ 2 ⊂ M 5 with geodesic edges γ 12 , γ 13 and η, where γ 23 is a minimal geodesic from p 2 to p 3 . Now, if we replace γ 13 with tγ 13 , where t ∈ S 1 , we obtain another flat, totally geodesic triangular surface △ 2 t ⊂ M 5 with geodesic edges γ 12 , tγ 13 and η t , where η t is a minimal geodesic from p 2 to tp 3 ∈ N 3 . In particular,…”