Integrability in Submanifold Geometry

Abstract

This thesis concerns the relationship of submanifold geometry, in both the smooth and discrete sense, to representation theory and the theory of integrable systems. We obtain Lie theoretic generalisations of the transformation theory of projectively and Lie applicable surfaces, and Mobius-flat submanifolds of the conformal n-sphere. In the former case, we propose a discretisation.We develop a projective approach to centro-ane hypersurfaces, analogous to theconformal approach to submanifolds in spaceforms. This yields a characterisation of centro-ane hypersurfaces amongst Mobius-flat projective hypersurfaces using polynomial conserved quantities.We also propose a discretisation of curved flats in symmetric spaces. After developing the transformation theory for this, we see how Darboux pairs of discrete isothermicnets arise as discrete curved flats in the symmetric space of opposite point pairs. We show how discrete curves in the 2-sphere fit into this framework.

Date of Award	30 Apr 2012
Original language	English
Awarding Institution	University of Bath
Supervisor	David Calderbank (Supervisor)