The toda equations and congruence in flag manifolds

Abstract

This thesis is concerned with the 2-dimensional Toda equations and their geometric interpretation in form of r-adapted maps into flag manifolds, r-adapted maps are not only of interest due to their relation with the Toda equations, but also for their adaption to the m-synametric space structure of flag manifolds. This thesis studies the congruence question for r-adapted maps in flag manifolds. The main theorem of this thesis is a congruence theorem for г-holomorphic maps Ψ : S(^2) → G/T of constant curvature, where G can be any compact simple Lie group. It is supplemented by a congruence theorem for general r-holomorphic maps Ψ : S(^2) → G/T if G has rank 2, and a number of congruence theorems for isometric r-primitive Ψ : S(^2) → G/T of constant Kahler angle. The second group of congruence theorems is proved for the rank 2 case, as well as a selection of Lie groups with higher rank: SU(4),SU(5),F(_4),E(_6),E(_6),E(_8),Sp(n).