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International Congress on Differential Geometry in memory of Alfred Gray (1939-1998)
September 18-23, 2000
Universidad del País Vasco
Bilbao, Spain

Organizers
M. Fernández (chairman), R. Ibáñez, M. Macho-Stadler

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Harmonic surfaces on flag manifolds
by
Caio Jose Colletti Negreiros
IMECC-UNICAMP, Brazil
Coauthors: Xiaohuan Mo (Peking University)

Oral communication

Lichnerowicz proved that if j:(M2, J1, g) --> (Nn, J2, h) is holomorphic and h is (1, 2)-sympletic then j is harmonic. Therefore, to understand about (1, 2)-sympletic metrics gives several examples of harmonic maps.

Burstall-Salamon showed that the study of harmonic maps into flag manifolds is naturally associated to tournaments. We can base ourselves by this viewpoint and give an alternative proof of the theorem of Gray-Wolf saying that the Cartan-Killing metric on F(n) is (1, 2)-sympletic iff n <= 3 (We use in a essential al way a theorem proved by Gale.)

Mo and I (to appear in Tohoku Mathematical Journal) have proved that if a Borel type metric in F(n) is (1, 2)-sympletic then the associated tournament \tau(J) contains only 4-tounaments isomorphic to those of two precise types. We use in a essential way the moving frame method in the proof of the results above.

Date received: March 2, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cadq-14.