Nihonkai Mathematical Journal

A sphere theorem for radial curvature

Nobuhiro Innami, Katsuhiro Shiohama, and Yuya Uneme

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We introduce a new constant for the surfaces of revolution homeomorphic to the 2-sphere. We prove a sphere theorem for radial curvature, assuming an inequality in the constant and the ratio of the difference of the maximal distance to the base point from the diameter of the reference surface and the injectivity radius of the base point. Namely, if a compact pointed Riemannian $n$-manifold which is referred to a surface of revolution satisfies the inequality, then it is topologically an $n$-sphere.

Article information

Nihonkai Math. J., Volume 24, Number 2 (2013), 93-102.

First available in Project Euclid: 24 February 2014

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53C20: Global Riemannian geometry, including pinching [See also 31C12, 58B20]
Secondary: 53C22: Geodesics [See also 58E10]

Toponogov comparison theorem sphere theorem radial curvature surface of revolution


Innami, Nobuhiro; Shiohama, Katsuhiro; Uneme, Yuya. A sphere theorem for radial curvature. Nihonkai Math. J. 24 (2013), no. 2, 93--102.

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