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

## TL;DR

This paper explores the geometry of  equipped with Warren's Ke4hler metric, demonstrating its flatness, deriving explicit geodesic and volume formulas, and constructing a family of similar flat metrics.

## Contribution

It provides a detailed geometric analysis of Warren's metric on , including explicit formulas and new flat metric constructions.

## Key findings

-  with Warren's metric is a flat manifold.
- Explicit formulas for geodesics and volume of geodesic balls are derived.
- A family of similar flat metrics is constructed.

## Abstract

The aim of this note is to describe the geometry of $\mathbb{C}^2$ equipped with a K\"{a}hler 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.