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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.

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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.
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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
  • 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