Topology of non-negatively curved manifolds

Abstract

An important question in the study of Riemannian manifolds of positive sectional curvature is how to distinguish manifolds that admit a metric with non-negative sectional curvature from those that admit one of positive curvature. Surprisingly, if the manifolds are compact and simply connected, all known obstructions to positive curvature are already obstructions to non-negative curvature. On the other hand, there are very few known examples of manifolds with positive curvature. They consist, apart from the rank one symmetric spaces, of certain homogeneous spaces G/H in dimensions 6, 7, 12, 13 Among the known examples of positive curvature there are two infinite families: in dimension 7 one has the homogeneous Aloff-Wallach spaces, and more generally the Eschenburg biquotients, and in dimension 13 the Bazaikin spaces. The topology of these manifolds has been studied extensively, see [KS1, KS2, AMP1, AMP2, Kr1, Kr2, Kr3, Sh, CEZ, FZ]. There exist many 7-dimensional positively curved examples which are homeomorphic to each other but not diffeomorphic, whereas in dimension 13, they are conjectured to be diffeomorphically distinct [FZ].In contrast to the positive curvature setting, there exist comparatively many examples with non-negative sectional curvature. The bi-invariant metric on a compact Lie group G induces, by O'Neill's formula, non-negative curvature on any homogeneous space G/H or more generally on any biquotient K\G/H. In [GZ1] a large new family of cohomogeneity one manifolds with non-negative curvature was constructed, giving rise to non-negatively curved metrics on exotic spheres. Hence it is natural to ask whether, among the known examples, it is possible to topologically distinguish manifolds with non-negative curvature from those admitting positive curvature. The purpose of this article is to address this question. There are many examples of non-negatively curved manifolds which are not homotopy equivalent to any of the known positively curved examples simply because they have different cohomology rings. But recently new families of non-negatively curved manifolds were discovered [GZ2] which, as we will see, give rise to several new manifolds having the same cohomology ring as the 7-dimensional Eschenburg spaces.Recall that the Eschenburg biquotients are defined as

Spherical T-duality II: An infinity of spherical T-duals for non-principal SU(2)-bundles

“…Then the main Theorem in [15] (with useful clarifications in [13,16]) shows that Lemma 2.3. Any principal SO(4)-bundle Q over a simply-connected, compact, oriented 4 dimensional manifold M is classified by the invariants…”

“…The Aloff-Wallach space W k,l [1], defined as being the homogeneous space SU(3)/T k,l , where the circle subgroup T k,l = diag(z k , z l , z −(k+l) ), |z| = 1, is a non-principal S 3 -bundle over CP 2 iff |k + l| = 1. We have W p,1−p ∼ = S −1,p(p−1) , where S p,q = E CP 2 (1, p, q), in our notation above, and e(S a,b ) = a − b, p 1 (S a,b ) = 2(a + b) + 1 [16]. Hence…”

Symplectic and Kähler structures on $$\mathbb CP^1$$-bundles over $$\mathbb CP^2$$

“…On the other hand, Eschenburg's manifold is diffeomorphic to the projectivisation of a rank 2 bundle over CP 2 [12,Theorem 2]. Furthermore the vector bundle can be taken with c 1 = 1, c 2 = − 1 [13]. Note finally, that a projectivisation of a bundle with c 1 = 1, c 2 = − 1 is diffeomorphic to the projectivisation of a bundle with c 1 = − 1, c 2 = − 1 since the latter is obtained from the former by tensoring with a line bundle.…”

GKM theory and Hamiltonian non-Kähler actions in dimension 6

“…It is known that the Eschenburg flag admits a Kähler structure; this is implicit in work of Eschenburg [7,Theorem 2] and Escher-Ziller [8], and explicit in [12,Section 4].…”

“…In this paper, we compare these examples to a third closely related example, namely a Hamiltonian T 2 -action on Eschenburg's twisted flag SU(3)//T 2 , constructed in [12]. We will show that both Tolman's and Woodward's example are (nonequivariantly) diffeomorphic to this manifold; as it is known [7,Theorem 2], [8], [12,Section 4] that the Eschenburg flag admits a Kähler structure we can answer also the question on the existence of a (noninvariant) Kähler structure on these examples in the affirmative.…”