Tohoku Mathematical Journal
2022
September
SECOND SERIES VOL. 74, NO. 3

Tohoku Math. J.
74 (2022), 329-364

Title NECESSARY AND SUFFICIENT CONDITIONS FOR A TRIANGLE COMPARISON THEOREM

Author James J. Hebda and Yutaka Ikeda

(Received May 26, 2020, revised December 9, 2020)
Abstract. We prove a version of Topogonov's triangle comparison theorem with surfaces of revolution as model spaces. Given a model surface and a Riemannian manifold with a fixed base point, we give necessary and sufficient conditions under which every geodesic triangle in the manifold with a vertex at the base point has a corresponding Alexandrov triangle in the model. Under these conditions we also prove a version of the Maximal Radius Theorem and a Grove--Shiohama type Sphere Theorem.

Mathematics Subject Classification. Primary 53C20; Secondary 53C22.
Key words and phrases. Generalized Toponogov triangle theorem, maximal radius theorem, Grove--Shiohama type theorem, cut locus, weaker radial attraction.

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