On questions of Pogorelov and Toponogov

Lei Ni, Wei Zhang, Yijian Zhang^* (corresponding author)

* Corresponding author: Yijian Zhang.

College Mathematics, Vol 1, Issue 1 (Winter 2026), pp. 1-12.

This article belongs to an issue that is still in progress. Bibliographic details may change before formal publication in December, 2026.

Abstract

We give explicit counterexamples to two questions. One is asked by Pogorelov and the other is by Toponogov, at least both are stated in. These questions concern the existence of closed asymptotic curves in a saddle surface, namely a complete immersed regular surface in 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.

How to cite

Lei Ni, Wei Zhang, Yijian Zhang. “On questions of Pogorelov and Toponogov.” College Mathematics 1, no. 1 (2026): 1-12.