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 r0 > 0 tangent disk bundle TM_r0 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 nonvanishing, then the Kähler manifolds TM_r0 have holonomy 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 TS^2 , giving us a new most natural description of this well-known metric.