Nihonkai Mathematical Journal

The Alexandrov-Toponogov comparison theorem for radial curvature

Nobuhiro Innami, Katsuhiro Shiohama, and Yuya Uneme

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We discuss the Alexandrov-Toponogov comparison theorem under the conditions of radial curvature of a pointed manifold $(M,o)$ with reference surface of revolution $(\widetilde M, \tilde o)$. There are two obstructions to make the comparison theorem for a triangle one of whose vertices is a base point $o$. One is the cut points of another vertex $\tilde p \not=\tilde o$ of a comparison triangle in $\widetilde M$. The other is the cut points of the base point $o$ in $M$. We find a condition under which the comparison theorem is valid for any geodesic triangle with a vertex at $o$ in $M$.

Article information

Nihonkai Math. J., Volume 24, Number 2 (2013), 57-91.

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 geodesic radial curvature surface of revolution


Innami, Nobuhiro; Shiohama, Katsuhiro; Uneme, Yuya. The Alexandrov-Toponogov comparison theorem for radial curvature. Nihonkai Math. J. 24 (2013), no. 2, 57--91.

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