Abstract
We study nonnegatively curved metrics on \({S^2\times\mathbb{R}^4}\). First, we prove rigidity theorems for connection metrics; for example, the holonomy group of the normal bundle of the soul must lie in a maximal torus of SO(4). Next, we prove that Wilking’s almost-positively curved metric on S 2 × S 3 extends to a nonnegatively curved metric on \({S^2\times\mathbb{R}^4}\) (so that Wilking’s space becomes the distance sphere of radius 1 about the soul). We describe in detail the geometry of this extended metric.
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References
Cheeger J., Gromoll D.: On the structure of complete manifolds of nonnegative curvature. Ann. of Math. 96, 413–443 (1972)
Gromoll D., Tapp K.: Nonnegatively curved metrics on \({S^2\times\mathbb{R}^2}\). Geom. Dedicata. 99, 127–136 (2003)
Guijarro L., Walschap G.: The metric projection onto the soul. Trans. Amer. Math. Soc. 352(1), 55–69 (2000)
Strake M., Walschap G.: Connection metrics of nonnegative curvature on vector bundles. Manuscripta Math. 66, 309–318 (1990)
Tapp, K.: Conditions for nonnegative curvature on vector bundles and sphere bundles. Duke Math. J. 116(1), 77–101 (2003)
Tapp K.: Quasi-positive curvature on homogeneous bundles. J. Differential Geom. 65, 273–287 (2003)
Tapp K.: Rigidity for nonnegatively curved metrics on \({S^2\times\mathbb{R}^3}\). Ann. Global Anal. Geom. 25, 43–58 (2004)
Wilking B.: Manifolds with positive sectional curvature almost everywhere. Invent. Math. 148, 117–141 (2002)
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Tapp, K. Metrics with nonnegative curvature on \({S^2 \times \mathbb{R}^4}\) . Ann Glob Anal Geom 42, 61–77 (2012). https://doi.org/10.1007/s10455-011-9301-1
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DOI: https://doi.org/10.1007/s10455-011-9301-1