Further Examples and Open Problems

Abstract

In this chapter we describe three Kahler manifolds with interesting properties. The first section summarizes the results in Loi and Zedda (J Geom Phys 110:269–276, 2016) showing that the complex plane \(\mathbb {C}\) endowed with the Cigar metric does not admit a local Kahler immersion into any complex space form even when the metric is rescaled by a positive constant. The importance of this example relies on the fact that there are not topological and geometrical obstructions for the existence of such an immersion. In the second section we describe a complete and not locally homogeneous metric introduced by Calabi (A construction of nonhomogeneous Einstein metrics. In: Proceedings of symposia in pure mathematics, vol 27, 1975). The diastasis function associated to this metric is not explicitly given and it makes very difficult to say something about the existence of a Kahler immersion into complex space forms . Finally, in the third and last section we discuss a 1-parameter family of nontrivial Ricci–flat metrics on \(\mathbb {C}^2\), called Taub-NUT metrics. The diastasis associated to these metrics is rotation invariant, i.e. depends only on the module of the variables, but it is not explicitly given and it is still unknown whether or not they are projectively induced for small values of the parameter.