×

Connection metrics of nonnegative curvature on vector bundles. (English) Zbl 0695.53032

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.
Reviewer: L.Tamássy

MSC:

53C20 Global Riemannian geometry, including pinching
53C05 Connections (general theory)
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] A. Besse, ”Einstein Manifolds,” Springer-Verlag, Berlin-Heidelberg, 1987 · Zbl 0613.53001
[2] F.R. Gantmacher, ”The Theory of Matrices”, Chelsea Publishing Co, New York, 1960 · Zbl 0088.25103
[3] H.B. Lawson, ”The Theory of Gauge Fields in Four Dimensions,” C.B.M.S. no 58, Amer. Math. Soc., 1985 · Zbl 0597.53001
[4] N. Steenrod, ”The Topology of Fibre Bundles”, Princeton University Press, Princeton, N.J., 1951 · Zbl 0054.07103
[5] M. Strake and G. Walschap,{\(\Sigma\)}-flat Manifolds and Riemannian Submersions, Manuscripta math. 64 (1989), 213–226 · Zbl 0687.53039 · doi:10.1007/BF01160120
[6] J. Vilms,Totally Geodesic Maps, J. Differential Geometry 4 (1970), 73–79 · Zbl 0194.52901
[7] G. Walschap,Nonnegatively Curved Manifolds with Souls of Codimension 2, J. Differential Geometry 27 (1988), 525–537 · Zbl 0654.53048
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.