Skip to Main Content

Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.


Compatible complex structures on almost quaternionic manifolds
HTML articles powered by AMS MathViewer

by D. V. Alekseevsky, S. Marchiafava and M. Pontecorvo PDF
Trans. Amer. Math. Soc. 351 (1999), 997-1014 Request permission


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.
  • 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
Similar Articles
  • Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 53C10, 32C10
  • Retrieve articles in all journals with MSC (1991): 53C10, 32C10
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:
  • S. Marchiafava
  • Affiliation: Dipartimento di Matematica, Università di Roma “La Sapienza", P.le A. Moro 2, 00185 Roma, Italy
  • Email:
  • M. Pontecorvo
  • Affiliation: Dipartimento di Matematica, Università di Roma Tre, L.go S.L. Murialdo 1, 00146 Roma, Italy
  • Email:
  • 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:
  • MathSciNet review: 1475674