Nihonkai Mathematical Journal

A sphere theorem for radial curvature

Nobuhiro Innami, Katsuhiro Shiohama, and Yuya Uneme

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Abstract

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

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

Dates
First available in Project Euclid: 24 February 2014

Permanent link to this document
https://projecteuclid.org/euclid.nihmj/1393273816

Mathematical Reviews number (MathSciNet)
MR3178501

Zentralblatt MATH identifier
1288.53027

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

Keywords
Toponogov comparison theorem sphere theorem radial curvature surface of revolution

Citation

Innami, Nobuhiro; Shiohama, Katsuhiro; Uneme, Yuya. A sphere theorem for radial curvature. Nihonkai Math. J. 24 (2013), no. 2, 93--102. https://projecteuclid.org/euclid.nihmj/1393273816


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