Gauss Metric on the Kummer Surface

TL;DR

This paper investigates two Gauss metrics on the Kummer surface, revealing they are not Ricci flat, and discusses the special case of the double sphere with its Einstein metric and non-vanishing first Chern class.

Contribution

It introduces two distinct Gauss metrics on the Kummer surface and analyzes their Ricci curvature properties, highlighting differences from Ricci flatness and examining the double sphere case.

Findings

The two Gauss metrics on the Kummer surface are not Ricci flat.

The double sphere has a Kähler and Einstein metric.

The first Chern class of the double sphere does not vanish.

Abstract

On the Kummer surface, we have obtained two different Gauss metrices by parametrizing it in two ways. We have found that these Gauss metrices are not Ricci flat. The double sphere, which is the special case of the Kummer surface, has the K\"{a}hler metric and the first Chern class of it does not vanish. Its metric is the Einstein metric which is not Ricci flat.