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Replacing the lower curvature bound in Toponogov’s comparison theorem by a weaker hypothesis. (English) Zbl 1376.53061
The following main theorem is proved.
Theorem. Let $$M$$ be a complete Riemannian manifold with a base point $$o$$, and let $$\tilde{M}$$ be a complete, simply connected surface which is rotationally symmetric about the vertex $$\tilde{o}$$ such that the cut locus of every point $$\tilde{p}$$ lies in the opposite meridian of $$\tilde{p}$$. Assume that $$(\tilde{M},\tilde{o})$$ has weaker radial attraction than $$(M,o)$$. Then, for every geodesic triangle $$\Delta opq$$ in $$M$$, there exists a geodesic triangle $$\Delta\tilde{o}\tilde{p}\tilde{q}$$ in $$\tilde{M}$$ whose corresponding sides are equal and which satisfies the angle comparison and Alexandrov convexity in Toponogov’s Theorem.
##### MSC:
 53C20 Global Riemannian geometry, including pinching 53C22 Geodesics in global differential geometry
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