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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Compatible complex structures
on almost quaternionic manifolds


Authors: D. V. Alekseevsky, S. Marchiafava and M. Pontecorvo
Journal: Trans. Amer. Math. Soc. 351 (1999), 997-1014
MSC (1991): Primary 53C10, 32C10
MathSciNet review: 1475674
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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.


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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
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

DOI: http://dx.doi.org/10.1090/S0002-9947-99-02201-1
PII: S 0002-9947(99)02201-1
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).
Article copyright: © Copyright 1999 American Mathematical Society