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arXiv · 2606.29231

On questions of Pogorelov and Toponogov

Lei Ni·Wei Zhang·Yijian Zhang
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Abstract

We give explicit counterexamples to two questions. One is asked by Pogorelov and the other is by Toponogov. These questions concern the existence of closed asymptotic curves in a saddle surface, namely a complete immersed regular surface in $\mathbb{R}^3$ with nonpositive Gaussian/sectional curvature, and its geometric consequences under some topological conditions. We also modify the statements and prove a corrected version. In the appendix we include an example clarifying a conjecture of Milnor.

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Lei Ni, Wei Zhang, Yijian Zhang. 2026-06-28. On questions of Pogorelov and Toponogov. https://arxiv.org/abs/2606.29231

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