arXiv subjects
Publications and source records attributed to Dirk Ferus.
1. Translated by Thomas E. Cecil, Department of Mathematics and Computer Science, College of the Holy Cross, Worcester, MA 01610, USA; E-mail address: cecil@mathcs.holycross.edu 2. Typed by Wenjiao Yan, School of Mathematical Sciences, Laboratory of Mathematics and Complex Systems, Beijing Normal University, Beijing 100875, China. E-mail address: wjyan@mail.bnu.edu.cn
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. Curved flats admit a 1-parameter family of deformations (spectral parameter) which enables us to make contact to integrable system theory. 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.
This paper studies isometric immersions of space forms by means of a hierarchy of finite dimensional integrable systems in Lax form on loop algebras.