The Harmonic Maps To A Class Of Symmetric Spaces And Their Applications

This paper mainly discusses harmonic maps to a class of symmetric spaces and their applications. The harmonic maps from a simply connected domainΩ(?)R²âˆªï½›âˆžï½into some symmetric spaces M_k, which can be embedded into some real forms G of unitary group U(N) and contain real Grassmann manifolds and Quaternion manifolds, are studied. The G-Grassmann extended solution is defined and some properties are presented. Then the G-Grassmann-uniton is introduced and constructed from a known one by the method of the dressing action, backlund transformation and flag transformation. It is proved that any harmonic mapφ:Ω→M_k with finite uniton number can be factorized into a product of a finite number of G-Grassmann- unitons which have the forms (Ï€+λπ^⊥)(Ï€_*+λ^-1Ï€_*^⊥). Furthermore, a way to construct a sequence of isotropic harmonic maps into G-Grassmann manifold is given according to some properties about the isotropic harmonic maps. By using the methods of explicit construction for uniton and G-uniton via adding unitons, the sufficient and necessary conditions for adding G-Grassmann- unitons are given considering the particularity of G-Grassmann manifold and the explicit construction of AUN-G-flag factor for commutative G -Grassmann-extended solution is given. Specially, the ways to explicitly construct S¹-invariant G extended solution are studied.