Connection metrics of nonnegative curvature on vector bundles
Let \(R^ k\to E\to^{\pi}M\) be a Euclidean vector bundle with Riemannian connection \(\nabla\) over a compact manifold M of positive sectional curvature. It is proved that if \(\nabla\) is radially symmetric, then E admits a complete metric g of nonnegative sectional curvature. Although radial symmetry is not necessary, a differential inequality is given which must be satisfied by the curvature tensor \(R^{\nabla}\) if (E,g) is nonnegatively curved. It is shown that a slightly sharper inequality is actually sufficient if \(k=2\), or \(R^{\nabla}\) is sufficiently nontrivial.
scientific article; zbMATH DE number 4138698