An invariant K\"ahler metric on the tangent disk bundle of a space-form

TL;DR

This paper constructs a family of invariant K"ahler metrics on tangent disk bundles of space-forms, revealing new Ricci-flat examples and connecting to known metrics like the Stenzel metric in specific cases.

Contribution

It introduces a new family of invariant K"ahler metrics on tangent disk bundles of space-forms, including Ricci-flat cases and a novel description of the Stenzel metric.

Findings

Metrics are complete for non-negative curvature

Metrics are non-complete for negative curvature

In 2D with constant curvature, metrics have SU(2) holonomy and are Ricci-flat

Abstract

We find a family of K\"ahler metrics invariantly defined on the radius $r_0>0$ tangent disk bundle ${{\cal T}_{M,r_0}}$ of any given real space-form $M$ or any of its quotients by discrete groups of isometries. Such metrics are complete in the non-negative curvature case and non-complete in the negative curvature case. If $\dim M=2$ and $M$ has constant sectional curvature $K\neq0$, then the K\"ahler manifolds ${{\cal T}_{M,r_0}}$ have holonomy $\mathrm{SU}(2)$; hence they are Ricci-flat. For $M=S^2$, just this dimension, the metric coincides with the Stenzel metric on the tangent manifold ${{\cal T}_{S^2}}$, giving us a new most natural description of this well-know metric.