Metrics of non-negative sectional curvature on $S^7$-bundles over $S^8$

Question

In Curvature and symmetry of Milnor spheres, Grove and Ziller construct metrics of non-negative sectional curvature on $S^3$-bundles over $S^4$ (by using a cohomogeneity one action). In the same paper, they ask whether this can be done in other dimensions (see Problem 5.1). Does anybody know whether there has been progress on this matter? I would particularily like to know if the total space of linear $S^7$-bundles over $S^8$ carry metrics of non-negative sectional curvature.

Answer 1

My understanding is that this is generally unknown. Of course, a few of the total spaces (e.g., $S^7\times S^8$, $S^{15}$, and the unit tangent bundle of $S^8$) are homogeneous spaces, so admit a non-negatively curved metric.

For the most interesting class of $S^7$ bundles over $S^8$ (the exotic $15$-dimensional spheres), more is known. First, none of these exotic spheres (regardless of whether or not they have the bundle structure) is a biquotient or a homogeneous space. This is a result of Totaro and independently by Kapovitch and Ziller.

Second, the cohomogeneity one approach of Grove and Ziller will not equip exotic $15$-spheres with non-negatively curved metrics. More generally, the generalization of this approach by Goette, Kerin, and Shankar (which recently established the existence of non-negatively curved metrics on all exotic $7$-spheres) cannot work on exotic $15$-spheres. The issue is that both approaches require the two singular leaves to both have codimension $2$. But Galaz-García, Kerin, and myself recently proved that a simply connected rational cohomology sphere $\Sigma^n$ has such a foliation only when $n\in \{2,3,5,7\}$.