Compatible complex structures on almost quaternionic manifolds
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- by D. V. Alekseevsky, S. Marchiafava and M. Pontecorvo PDF
- Trans. Amer. Math. Soc. 351 (1999), 997-1014 Request permission
Abstract:
On an almost quaternionic manifold $(M^{4n}, Q)$ we study the integrability of almost complex structures which are compatible with the almost quaternionic structure $Q$. If $n\geq 2$, we prove that the existence of two compatible complex structures $I_{1}, I_{2}\neq \pm I_{1}$ forces $(M^{4n}, Q)$ to be quaternionic. If $n=1$, that is $(M^{4},Q)=(M^{4},[g],or)$ is an oriented conformal 4-manifold, we prove a maximum principle for the angle function $\langle I_{1},I_{2}\rangle$ of two compatible complex structures and deduce an application to anti-self-dual manifolds. By considering the special class of Oproiu connections we prove the existence of a well defined almost complex structure $\mathbb {J}$ on the twistor space $Z$ of an almost quaternionic manifold $(M^{4n}, Q)$ and show that $\mathbb {J}$ is a complex structure if and only if $Q$ is quaternionic. This is a natural generalization of the Penrose twistor constructions.References
- E. Abbena, S. Garbiero, S. Salamon, Hermitian geometry on the Iwasawa manifold, Preprint 1995.
- D. V. Alekseevsky and M. M. Graev, $G$-structures of twistor type and their twistor spaces, J. Geom. Phys. 10 (1993), no. 3, 203–229. MR 1215608, DOI 10.1016/0393-0440(93)90015-7
- 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. MR 506229, DOI 10.1098/rspa.1978.0143
- D.V. Alekseevsky, S. Marchiafava, Quaternionic structures on a manifold and subordinated structures, Annali di Mat. Pura e Appl. (4) 171 (1996), 205-273.
- 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. MR 867684, DOI 10.1007/978-3-540-74311-8
- Charles P. Boyer, A note on hyper-Hermitian four-manifolds, Proc. Amer. Math. Soc. 102 (1988), no. 1, 157–164. MR 915736, DOI 10.1090/S0002-9939-1988-0915736-8
- D. Burns and P. De Bartolomeis, Applications harmoniques stables dans $\textbf {P}^n$, Ann. Sci. École Norm. Sup. (4) 21 (1988), no. 2, 159–177 (French). MR 956764
- 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). MR 1153706
- 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. MR 1365975
- —, Canonical connections for almost-hypercomplex structures, Pitman Res. Notes in Math. Ser., Longman, Harlow, 1997.
- G. Grantcharov, Private communications.
- P. Kobak, Explicit doubly-Hermitian metrics, ESI preprint (1995).
- 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. MR 0238225
- Claude LeBrun, Quaternionic-Kähler manifolds and conformal geometry, Math. Ann. 284 (1989), no. 3, 353–376. MR 1001707, DOI 10.1007/BF01442490
- Vasile Oproiu, Integrability of almost quaternal structures, An. Ştiinţ. Univ. Al. I. Cuza Iaşi Secţ. I a Mat. 30 (1984), no. 5, 75–84. MR 800155
- Massimiliano Pontecorvo, Complex structures on quaternionic manifolds, Differential Geom. Appl. 4 (1994), no. 2, 163–177. MR 1279015, DOI 10.1016/0926-2245(94)00012-3
- —, Complex structures on Riemannian $4$-manifolds, Math. Ann. 309 (1997), 159–177.
- 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. MR 1040847
- S. M. Salamon, Quaternionic manifolds, Symposia Mathematica, Vol. XXVI (Rome, 1980) Academic Press, London-New York, 1982, pp. 139–151. MR 663029
- Simon Salamon, Harmonic and holomorphic maps, Geometry seminar “Luigi Bianchi” II—1984, Lecture Notes in Math., vol. 1164, Springer, Berlin, 1985, pp. 161–224. MR 829230, DOI 10.1007/BFb0081912
- S. M. Salamon, Differential geometry of quaternionic manifolds, Ann. Sci. École Norm. Sup. (4) 19 (1986), no. 1, 31–55. MR 860810
- 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). MR 1219803
- Franco Tricerri, Sulle varietà dotate di due strutture quasi complesse linearmente indipendenti, Riv. Mat. Univ. Parma (3) 3 (1974), 349–358. MR 431032
- 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). MR 442866
Additional Information
- D. V. Alekseevsky
- Affiliation: Gen. Antonova 2, kv. 99, 117279 Moscow, Russian Federation
- Address at time of publication: E. Schrödinger Institute, Bolzmanngasse 9, A-1090, Vienna, Austria
- MR Author ID: 226278
- ORCID: 0000-0002-6622-7975
- Email: daleksee@esi.ac.at
- S. Marchiafava
- Affiliation: Dipartimento di Matematica, Università di Roma “La Sapienza", P.le A. Moro 2, 00185 Roma, Italy
- Email: marchiafava@axrma.uniroma1.it
- M. Pontecorvo
- Affiliation: Dipartimento di Matematica, Università di Roma Tre, L.go S.L. Murialdo 1, 00146 Roma, Italy
- Email: max@matrm3.mat.uniroma3.it
- Received by editor(s): December 14, 1996
- Additional Notes: Work done under the program of G.N.S.A.G.A. of C.N.R. and partially supported by M.U.R.S.T. (Italy) and E.S.I. (Vienna).
- © Copyright 1999 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 351 (1999), 997-1014
- MSC (1991): Primary 53C10, 32C10
- DOI: https://doi.org/10.1090/S0002-9947-99-02201-1
- MathSciNet review: 1475674