Half plane geometries

TL;DR

This paper explores warped product metrics on the right half plane, analyzing their isometry groups and curvature properties, revealing new geometric structures with specific curvature behaviors and incompleteness.

Contribution

It introduces specific warped product metrics on the right half plane and characterizes their isometry groups and curvature properties, highlighting novel geometric features.

Findings

Isometry group is isomorphic to the real additive group.

Identifies metrics with zero and unbounded negative sectional curvature.

Both metrics are not complete.

Abstract

In this paper, we endow the right half plane with warped product metrics. The group of holomorphic isometries of all such metrics is isomorphic to the real additive group. Of our interest are two of those metrics: they have zero and unbounded negative sectional curvature, respectively, and both of them are not complete.