Sfb 288 Differential Geometry and Quantum Physics

Abstract for Sfb Preprint No. 174

Curved flats in symmetric spaces

D. Ferus, F. Pedit

In this paper we study maps (curved flats) into symmetric spaces which are tangent at each point to a flat of the symmetric space. Important examples of such maps arise from isometric immersions of space forms into space forms via their Gauss maps. Further examples are found in conformal geometry, e.g. the curved flats obtained from isothermic surfaces and conformally flat 3-folds in the 4-sphere. Besides their geometric interest curved flats admit a 1-parameter family of deformations (spectral parameter) which enables us to make contact to integrable systems on loop algebras. In fact, we give a recipe to construct curved flats (and thus the above mentioned geometric objects) from a hierarchy of finite dimensional algebraically completely integrable flows.

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