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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.


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          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?


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              Problem 5.3.

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


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                  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.