Black, Malcolm (1990) Harmonic maps into homogenous spaces. PhD thesis, University of Warwick.

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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.

Item Type:	Thesis (PhD)
Subjects:	Q Science > QA Mathematics
Library of Congress Subject Headings (LCSH):	Harmonic maps, Homogeneous spaces
Official Date:	August 1990
Dates:	Date Event August 1990 Submitted
Institution:	University of Warwick
Theses Department:	Mathematics Institute
Thesis Type:	PhD
Publication Status:	Unpublished
Supervisor(s)/Advisor:	Rawnsley, John H. (John Howard), 1947-
Sponsors:	Science and Engineering Research Council (Great Britain)
Format of File:	pdf
Extent:	94 leaves
Language:	eng

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