A class of Osserman spaces

We prove that a symmetric space is Osserman if its complexification is a complex hyper-Kähler symmetric space. This includes all pseudo-hyper-Kähler as well as para-hyper-Kähler symmetric spaces. We extend the classification of pseudo-hyper-Kähler symmetric spaces obtained by the first and the third author to the class of para-hyper-Kähler symmetric spaces. These manifolds are possible targets for the scalars of rigid N=2 supersymmetric field theories with hypermultiplets on four-dimensional space-times with Euclidean signature.