An invariant Kähler metric on the tangent disk bundle of a space-form

Abstract

We find a family of Kähler metrics invariantly defined on the radius $r_0>0$ tangent disk bundle ${\mathcal{T}}_{M,r_0}$ 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 $K\ne 0$, then the Kähler manifolds ${\mathcal{T}}_{M,r_0}$ 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 ${\mathcal{T}}_{S^2}$, giving us a new most natural description of this well-know metric.

Further improvements, 14 pages. Accepted in Revue Roumaine de Mathématiques Pures et Appliquées

References in corpus (2)

• Homotheties and topology of tangent sphere bundles
• Weighted metrics on tangent sphere bundles