Soliton Theory Of Harmonic Maps And Isometric Immersions And Their Darboux Transformations

This paper is to study harmonic maps into symplectic groups and local isometric immersions into space forms by means of the soliton theory. By realizing an action of the rational loop group on the spaces of corrsponding solutions, we get the Backlund transformation and the Darboux transformation, and thereby we give the explicit construction for harmonic maps into symplectic groups and local isometric immersions into space forms via purely algebraic algorithm. This paper contains four chapters. Its content may be separated into two parts. The first part contains Chapter one and Chapter two, which treat of the harmonic maps from surface into symplectic groups and quaternion Grassmann manifolds. The second part contains Chapter three and Chapter four, which treat of local isometric immersions from space forms or Riemannian products of space forms into space forms.In Chapter one, by means of the theory of harmonic maps into the unitary group U(2N) ([U]), we study harmonic maps from a simply-connected domain into the symplectic group Sp(N). The symplectic uniton and symplectic extended uniton are introduced. The method of the symplectic Backlund transformation and the Darboux transformation is used to construct new symplectic uniton from a known one. Being based on this, we obtain the best estimate for minimal symplectic uniton number.LetSn(G)={all the symplectic extended n-unitons}, where G=GL(2N,C). meromorphic with no poles at (0,)Theorem 1.2.1 Given we have , and there is a unique The simplest element o is it is a Hermitian projection of C2N.Proposition 1.2.2 if and only if . In other words, there are no nontrivial simplest elements in.Lemma1.2.3 By using symplectic simplest element we give the Backlundtransformation and and Darboux transformation on Sn(G) (see Proposition 1 .2.8 and Theorem 1.2.9). If and are Hermitian projections of C2N such that is the extendedsolution of is the symplectic extended solutionof , we call a symplectic flag factor of , for simplicity,denoted by Proposition 1.3.6 Let be a symplectic n-uniton with the symplecticextended solution and rank(T-n)=k. Then we can construct a symplecticflag factor of rank(2N-r)for 0