Summary: Annals of Global Analysis and Geometry 16: 291308, 1998.
© 1998 Kluwer Academic Publishers. Printed in the Netherlands.
Hermitian Structures on Twistor Spaces
Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. G. Bonchev Str.,
bl.8, Sofia 1113, Bulgaria and Centre de Mathématiques, Ecole Polytechnique, CNRS URA 169,
91128 Palaiseau, France
G. GRANTCHAROV and S. IVANOV
Department of Geometry, Faculty of Mathematics and Informatics, University of Sofia, Blvd. James
Bouchier 5, Sofia 1164, Bulgaria
Abstract. The paper contains description of the orthogonal complex structures with respect to the
natural 1-parameter family of Riemannian metrics on the (negative) twistor space over a self-dual
Einstein Riemannian 4-manifold. We prove that if the twistor space of a compact self-dual Einstein
4-manifold admits more than one orthogonal complex structure then the 4-manifold has a Kähler
structure. Considering the flag manifold F1,2 which is the twistor space of CP2 endowed with the
FubiniStudy metric, we obtain that any invariant Einstein metric on F1,2 admits even locally exactly
three orthogonal complex structures which are the invariant ones.