A Riemannian plane with only two injective Geodesics

TL;DR

This paper constructs a unique example of a complete Riemannian plane that admits exactly two injective geodesics, highlighting a rare geometric property and exploring related open problems.

Contribution

It introduces a novel example of a Riemannian plane with only two injective geodesics, expanding understanding of geodesic behavior in differential geometry.

Findings

The constructed plane has exactly two injective geodesics.

The example is a perturbation of a surface of revolution with a contracting end.

Open problems related to geodesic properties are discussed.

Abstract

We present an example of a complete Riemannian plane with precisely two injective geodesics - up to reparameterization. The example arises as a perturbation of a surface of revolution with contracting end. The last section is devoted to open problems.