A Riemannian plane with only two injective geodesics (Q6624300)

Let \(M\) be a complete, noncompact Riemannian manifold. The geodesics on \(M\) can be divided into three classes. A geodesic \(\gamma: \mathbb{R} \to M\) is called proper if it is a proper map from \(\mathbb{R}\) to \(M\); or \(\gamma\) is called bounded, if \(\gamma(\mathbb{R})\) is a bounded subset of \(M\); otherwise \(\gamma\) is called oscillating. Moreover, a geodesic \(\gamma\) in \(M\) is called embedded, if the map \(\gamma: \mathbb{R} \to M\) is proper and injective. What is the largest number \(n\) such that every complete Riemannian plane has at least \(n\) embedded geodesics (up to reparameterization)?\N\NIt was previously known that \(n \ge 1\). The main result of the present paper is that there exist complete Riemannian planes with precisely two embedded geodesics. Therefore, \(n \le 2\). The authors present a conjecture: every complete Riemannian plane has at least two embedded geodesics.