Rigidity for odd-dimensional souls

Tapp, Kristopher¹

¹ Department of Mathematics, Saint Joseph’s University, 5600 City Avenue, Philadelphia PA 19131, USA

Geometry & topology, Tome 16 (2012) no. 2, pp. 957-962.

Résumé

We prove a new rigidity result for an open manifold $M$ with nonnegative sectional curvature whose soul $\Sigma \subset M$ is odd-dimensional. Specifically, there exists a geodesic in $\Sigma$ and a parallel vertical plane field along it with constant vertical curvature and vanishing normal curvature. Under the added assumption that the Sharafutdinov fibers are rotationally symmetric, this implies that for small $r$, the distance sphere $B_r\left(\Sigma \right)=\left\{p\in M\mid dist\left(p,\Sigma \right)=r\right\}$ contains an immersed flat cylinder, and thus could not have positive curvature.

Détail

DOI : 10.2140/gt.2012.16.957

Classification : 53C20
Keywords: Soul Theorem, nonnegative curvature, flat

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