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Compatible complex structures on almost quaternionic manifolds. (English) Zbl 0933.53017
Let \((M^{4n},Q)\) be an almost quaternionic manifold. Assume that \(Q\) is locally spanned by the almost complex structures \(J_1,J_1,J_3\) such that \(J_1J_2=-J_2J_1=J_3\). An almost complex structure \(I\) on \(M^{4n}\) is compatible with the almost quaternionic structure \(Q\) if \(I=\sum^2_{\alpha=1} c_\alpha J_\alpha\), locally, and \(\sum^3_{\alpha=1} c^2_\alpha=1\).
The authors study the relationship between the 1-integrability of the almost quaternionic structure \(Q\) and the existence of compatible complex structures. They obtain several very interesting results concerning the properties of the twistor space of an almost quaternionic manifold as well as some generalizations of the Penrose twistor constructions. In the case \(n\geq 2\), there are two compatible complex structures \(I_1,I_2 \neq\pm I_1\) on \(M^{4n}\) if \((M^{4n},Q)\) is quaternionic. If \(n=1\), a maximum principle for the angle function of two compatible almost complex structures follows. Then, an application to anti-self-dual manifolds is presented. There exists an almost complex structure on the twistor space \(Z\) of an almost quaternionic manifold \((M^{4n},Q)\), and it is integrable if and only if \(Q\) is quaternionic.
Reviewer: V.Oproiu (Iaşi)

MSC:
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
53C10 \(G\)-structures
53C26 Hyper-Kähler and quaternionic Kähler geometry, “special” geometry
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[1] E. Abbena, S. Garbiero, S. Salamon, Hermitian geometry on the Iwasawa manifold, Preprint 1995. · Zbl 0886.53046
[2] D. V. Alekseevsky and M. M. Graev, \?-structures of twistor type and their twistor spaces, J. Geom. Phys. 10 (1993), no. 3, 203 – 229. · Zbl 0779.53020
[3] M. F. Atiyah, N. J. Hitchin, and I. M. Singer, Self-duality in four-dimensional Riemannian geometry, Proc. Roy. Soc. London Ser. A 362 (1978), no. 1711, 425 – 461. · Zbl 0389.53011
[4] D.V. Alekseevsky, S. Marchiafava, Quaternionic structures on a manifold and subordinated structures, Annali di Mat. Pura e Appl. (4) 171 (1996), 205-273. CMP 97:10 · Zbl 0968.53033
[5] Arthur L. Besse, Einstein manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 10, Springer-Verlag, Berlin, 1987. · Zbl 0613.53001
[6] Charles P. Boyer, A note on hyper-Hermitian four-manifolds, Proc. Amer. Math. Soc. 102 (1988), no. 1, 157 – 164. · Zbl 0642.53073
[7] D. Burns and P. De Bartolomeis, Applications harmoniques stables dans \?\(^{n}\), Ann. Sci. École Norm. Sup. (4) 21 (1988), no. 2, 159 – 177 (French). · Zbl 0661.32035
[8] Paul Gauduchon, Structures de Weyl et théorèmes d’annulation sur une variété conforme autoduale, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 18 (1991), no. 4, 563 – 629 (French). · Zbl 0763.53034
[9] Paul Gauduchon, Complex structures on compact conformal manifolds of negative type, Complex analysis and geometry (Trento, 1993) Lecture Notes in Pure and Appl. Math., vol. 173, Dekker, New York, 1996, pp. 201 – 212. · Zbl 0870.53029
[10] -, Canonical connections for almost-hypercomplex structures, Pitman Res. Notes in Math. Ser., Longman, Harlow, 1997. CMP 98:03 · Zbl 0893.53014
[11] G. Grantcharov, Private communications.
[12] P. Kobak, Explicit doubly-Hermitian metrics, ESI preprint (1995). · Zbl 0947.53011
[13] Shoshichi Kobayashi and Katsumi Nomizu, Foundations of differential geometry. Vol. II, Interscience Tracts in Pure and Applied Mathematics, No. 15 Vol. II, Interscience Publishers John Wiley & Sons, Inc., New York-London-Sydney, 1969. · Zbl 0091.34802
[14] Claude LeBrun, Quaternionic-Kähler manifolds and conformal geometry, Math. Ann. 284 (1989), no. 3, 353 – 376. · Zbl 0674.53036
[15] Vasile Oproiu, Integrability of almost quaternal structures, An. Ştiinţ. Univ. Al. I. Cuza Iaşi Secţ. I a Mat. 30 (1984), no. 5, 75 – 84. · Zbl 0573.53021
[16] Massimiliano Pontecorvo, Complex structures on quaternionic manifolds, Differential Geom. Appl. 4 (1994), no. 2, 163 – 177. · Zbl 0797.53037
[17] -, Complex structures on Riemannian \(4\)-manifolds, Math. Ann. 309 (1997), 159-177. CMP 97:17
[18] Henrik Pedersen and Y. Sun Poon, Twistorial construction of quaternionic manifolds, Proceedings of the Sixth International Colloquium on Differential Geometry (Santiago de Compostela, 1988) Cursos Congr. Univ. Santiago de Compostela, vol. 61, Univ. Santiago de Compostela, Santiago de Compostela, 1989, pp. 207 – 218. · Zbl 0691.53053
[19] S. M. Salamon, Quaternionic manifolds, Symposia Mathematica, Vol. XXVI (Rome, 1980) Academic Press, London-New York, 1982, pp. 139 – 151.
[20] Simon Salamon, Harmonic and holomorphic maps, Geometry seminar ”Luigi Bianchi” II — 1984, Lecture Notes in Math., vol. 1164, Springer, Berlin, 1985, pp. 161 – 224.
[21] S. M. Salamon, Differential geometry of quaternionic manifolds, Ann. Sci. École Norm. Sup. (4) 19 (1986), no. 1, 31 – 55. · Zbl 0616.53023
[22] Simon Salamon, Special structures on four-manifolds, Riv. Mat. Univ. Parma (4) 17* (1991), 109 – 123 (1993). Conference on Differential Geometry and Topology (Italian) (Parma, 1991). · Zbl 0796.53031
[23] Franco Tricerri, Sulle varietà dotate di due strutture quasi complesse linearmente indipendenti, Riv. Mat. Univ. Parma (3) 3 (1974), 349 – 358. · Zbl 0344.53022
[24] Franco Tricerri, Connessioni lineari e metriche hermitiane sopra varietà dotate di due strutture quasi complesse, Riv. Mat. Univ. Parma (4) 1 (1975), 177 – 186 (Italian). · Zbl 0352.53016
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