5. Vector Bundles
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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.