Orbifold fibrations of Eschenburg spaces

Abstract

Most of the few known examples of compact Riemannian manifolds with positive sectional curvature are the total space of a Riemannian submersion. In this article we show that this is true for all known examples, if we enlarge the category to orbifold fibrations. For this purpose we study all almost free isometric circle actions on positively curved Eschenburg spaces, which give rise to principle orbifold bundle structures, and we examine in detail their geometric properties. In particular, we obtain a new family of 6-dimensional orbifolds with positive sectional curvature whose singular locus consists of just two points.

Orbifold biquotients of SU(3)

“…Another interesting property of this orbifold is that it admits an isometric S 1 action, and we will see that In this paper we also further study the orbifold quotients of Eschenburg spaces in a slightly more general way than done in [12]. We also provide some minor corrections and improvements, see Theorem 6.10.…”

“…Before examining the idea of orbifold fibrations of Eschenburg spaces by Florit and Ziller [12], let us first examine the orbifold structure of Eschenburg orbifolds, since it will be similar to that of the orbifold fibrations.…”

“…For the construction of the six dimensional family of Exchenburg spaces, Florit and Ziller [12] considered fibrations of the form…”

“…By conjugation, we may assume without loss of generality that σ = Id, τ = (12), so L ij = L 33 , and U (2) ij is the standard embedding of U (2) into SU (3). Now assume that (z, w) acts trivially on T 2 Id and T 2 (12) . Since I ∈ T 2 Id , and (z, w) X = diag(u 1 , (12) .…”

Topological obstructions to fatness

“…All of them, apart from some rank one symmetric spaces, can be viewed as the total space of a Riemannian submersion, in some cases an orbifold submersion; see Florit and Ziller [7]. The fact that the homogeneous ones also have totally geodesic fibers motivated A Weinstein [19] to study Riemannian submersions with totally geodesic fibers and positive vertizontal curvatures, ie, sectional curvatures of planes spanned by a vector tangent to a fiber and a vector orthogonal to it.…”