Sigma models, minimal surfaces and some Ricci flat pseudo-Riemannian geometries. (English) Zbl 1070.53503

Mladenov, Ivaïlo M. (ed.) et al., Proceedings of the 2nd international conference on geometry, integrability and quantization, Varna, Bulgaria, June 7–15, 2000. Sofia: Coral Press Scientific Publishing (ISBN 954-90618-2-5/pbk). 171-180 (2001).
Summary: 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.
For the entire collection see [Zbl 0957.00038].


53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53B30 Local differential geometry of Lorentz metrics, indefinite metrics
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