Sigma Models, Minimal Surfaces and Some Ricci Flat Pseudo-Riemannian Geometries

Abstract

We consider the sigma models where the base metric is proportional to the metric of the configuration space. We show that the corresponding sigma model equation admits a Lax pair. We also show that this type of sigma models in two dimensions are intimately related to the minimal surfaces in a flat pseudo-Riemannian 3-space. We define two dimensional surfaces conformally related to the minimal surfaces in flat three dimensional geometries which enable us to give a construction of the metrics of some even dimensional Ricci flat (pseudo-) Riemannian geometries.