The geometry of ${\mathbb{C}}^2$ equipped with Warren's metric

Abstract

The aim of this note is to describe the geometry of ${\mathbb{C}}^2$ equipped with a Kähler metric defined by Warren. It is shown that with that metric ${\mathbb{C}}^2$ is a flat manifold. Explicit formulae for geodesics and volume of geodesic ball are also computed. Finally, a family of similar flat metrics is constructed.

arXiv commentFinal version to appear in Annales Polonici Mathematici