Detecting quaternionic maps between hyperkähler manifolds. (English) Zbl 1308.53072

Summary: For a harmonic map between two hyper-Kähler manifolds, we prove a Weitzenböck type formula for the defining quantity of quaternionic maps, and apply it to harmonic morphisms. We also provide a sufficient and necessary condition for a smooth map being quaternionic.


53C26 Hyper-Kähler and quaternionic Kähler geometry, “special” geometry
53C43 Differential geometric aspects of harmonic maps
Full Text: DOI


[1] Berger, M.: Sur les groupes d’holonomie des variété à connexion affine et variété riemanniennes. Bull. Soc. Math. Fr. 83, 279-330 (1955) · Zbl 0068.36002
[2] Chen, J., Complex anti-self-dual connections on product of Calabi-Yau surfaces and triholomorphic curves, Commun. Math. Phys., 201, 201-247, (1999) · Zbl 0948.32027
[3] Chen, J.; Li, J., Quaternionic maps between hyperkähler manifolds, J. Differ. Geom., 55, 355-384, (2000) · Zbl 1067.53035
[4] Cheng, S.Y., Liouville theorem for harmonic maps, Proc. Symp. Pure Math., 36, 147-151, (1980)
[5] Eells, J., Lemaire, L.: Selected Topics in Harmonic Maps. C.B.M.S. Regional Conf. Series, vol. 50. Am. Math. Soc., Providence (1983) · Zbl 0515.58011
[6] Figuroa-O’Farrill, J.M.; Köhl, C.; Spence, B., Supersymmetric Yang-Mills, octonionic instantons and triholomorphic curves, Nucl. Phys. B, 521, 419-443, (1998) · Zbl 0954.53019
[7] Fuglede, B., Harmonic morphisms between Riemannian manifolds, Ann. Inst. Fourier (Grenoble), 28, 107-144, (1978) · Zbl 0339.53026
[8] Ishihara, T., A mapping of Riemannian manifolds which preserves harmonic functions, J. Math. Kyoto Univ., 19, 215-229, (1979) · Zbl 0421.31006
[9] Li, J.; Tian, G., A blow-up formula for stationary harmonic maps, Int. Math. Res. Not., 14, 735-755, (1998) · Zbl 0944.58010
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.