Harmonic maps into homogeneous spaces

Abstract

This thesis investigates harmonic maps into homogeneous spaces, principally flag manifolds. First we show that an f-holomorphic map of a Hermitian cosymplectic manifold is harmonic provided that the f-structure on the co-domain satisfies (d∇F)1,1 = 0, where ∇ is the Levi-Civita connection. We then characterize those invariant f-structures and metrics on homogeneous spaces which satisfy this condition. On a homogeneous space whose tangent bundle splits as a direct sum of mutually distinct isotropy spaces (e.g. a flag manifold), we see that an f-structure which is horizontal (i.e. [F+, F] ⊂ h) satisfies (d∇F)1,1 = 0 for any choice of invariant metric. Thus f-holomorphic maps are equi-harmonic (harmonic with respect to all invariant metrics). Equi-harmonic maps are seen to behave well in combination with homogeneous geometry. Next we classify horizontal f-structures on flag manifolds. The classification provides a unified framework for producing examples of flag manifolds fibring twistorially over homogeneous spaces. Another application of this classification is to find f-holomorphic orbits in full flag manifolds. Finally, we show that an equi-harmonic map of a Riemann surface which is also equi-weakly conformal is f-holomorphic with respect to a horizontal f-structure. Our classification theorem then allows a more concrete description of such maps bringing further examples to light.