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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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