A twistor description of harmonic maps of a 2-sphere into a Grassmannian
A twistor fibration \(\pi\) : \(Z\to N\) over a Riemannian manifold N is a Riemannian submersion of an almost Hermitian manifold Z onto N with the following property: If M is an almost Hermitian manifold with co-closed Kähler form and \(\psi\) : \(M\to Z\) is a holomorphic map then \(\pi_ 0\psi: M\to N\) is a harmonic map. In this article, certain natural twistor fibrations of complex flag manifolds over complex Grassmannians are exhibited. It is then shown that all harmonic maps of a Riemann sphere into a Grassmannian arise as projections of holomorphic maps into a flag manifold. Further, it is demonstrated that from every harmonic map of \(S^ 2\) into a Grassmannian, there may be constructed a family of transformed maps of \(S^ 2\) into other Grassmannians which are also harmonic.
complex flag manifolds over complex Grassmannians
scientific article; zbMATH DE number 3919405
scientific article