
## 5. Vector Bundles

1. #### Problem 5.1.

Is there a metric with $\sec \geq 0$ on $\R^6$-bundles over $S^3 \times S^3$ with non-trivial Euler class? Wilking has shown that the answer is “No” if the soul is $S^3 \times S^3$ with the product metric.
• #### Problem 5.2.

Which vector bundles over $S^2 \times S^2$ or $\C P^2 \# \pm \C P^2$ where the structure group does not reduce to a torus admit $\sec \geq 0$?
• #### Problem 5.3.

Classify metrics with $\sec \geq 0$ on $S^2 \times \R^4$ (or, more generally, on $S^n \times \R^k$).
• #### Problem 5.4.

Suppose $E \longrightarrow M$ is a vector bundle over a compact, simply-connected manifold $M$ for which $\sec_M \geq 0$. Does $E \oplus \R^k \longrightarrow M$ have $\sec \geq 0$ for $k$ large?

Cite this as: AimPL: Manifolds with non-negative sectional curvature, available at http://aimpl.org/nnsectcurvature.