5. Vector Bundles

Problem 5.1.
Is there a metric with $\sec \geq 0$ on $\R^6$bundles over $S^3 \times S^3$ with nontrivial 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, simplyconnected 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 nonnegative sectional curvature, available at http://aimpl.org/nnsectcurvature.