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# On twistor almost complex structures

• * Corresponding author: Simone Gutt

Dedicated to our friend Kirill Mackenzie

• In this paper we look at the question of integrability, or not, of the two natural almost complex structures $J^{\pm}_\nabla$ defined on the twistor space $J(M, g)$ of an even-dimensional manifold $M$ with additional structures $g$ and $\nabla$ a $g$-connection. We measure their non-integrability by the dimension of the span of the values of $N^{J^\pm_\nabla}$. We also look at the question of the compatibility of $J^{\pm}_\nabla$ with a natural closed $2$-form $\omega^{J(M, g, \nabla)}$ defined on $J(M, g)$. For $(M, g)$ we consider either a pseudo-Riemannian manifold, orientable or not, with the Levi Civita connection or a symplectic manifold with a given symplectic connection $\nabla$. In all cases $J(M, g)$ is a bundle of complex structures on the tangent spaces of $M$ compatible with $g$. In the case $M$ is oriented we require the orientation of the complex structures to be the given one. In the symplectic case the complex structures are positive.

Mathematics Subject Classification: Primary 53C15, 53C28; Secondary 53D99.

 Citation:

•  [1] M. F. Atiyah, N. J. Hitchin and I. M. Singer, Self-duality in four-dimensional Riemannian geometry, Proc. Roy. Soc. A, 362 (1978), 425-461.  doi: 10.1098/rspa.1978.0143. [2] M. Berger, Sur quelques variétés riemaniennes suffisamment pincées, Bulletin de la S.M.F., 88 (1960), 57-71. [3] A. L. Besse, Einstein Manifolds, Springer-Verlag, Berlin–Heidelberg, 1987. doi: 10.1007/978-3-540-74311-8. [4] M. Cahen, M. Gérard, S. Gutt and M. Hayyani, Distributions associated to almost complex structures on symplectic manifolds, preprint, arXiv: 2002.02335. [5] J. Eells and S. Salamon, Twistorial construction of harmonic maps of surfaces into four-manifolds, Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 4, 12 (1985), 589-640. [6] J. Fine and D. Panov, Symplectic Calabi-Yau manifolds, minimal surfaces and the hyperbolic geometry of the conifold, J. of Differential Geom., 82 (2009), 155-205. [7] J. Fine and D. Panov, Hyperbolic geometry and non-Kähler manifolds with trivial canonical bundle, Geometry and Topology, 14 (2010), 1723-1763. [8] N. R. O'Brian and J. H. Rawnsley, Twistor spaces, Annals of Global Analysis and Geometry, 3 (1985), 29-58.  doi: 10.1007/BF00054490. [9] J. H. Rawnsley, f-structures, f-twistor spaces and harmonic maps, in Geometry Seminar "Luigi Bianchi" II — 1984, 85–159, Lecture Notes in Math., 1164, Springer, Berlin, 1985. doi: 10.1007/BFb0081911. [10] A. G. Reznikov, Symplectic twistor spaces, Annals of Global Analysis and Geometry, 11 (1993), 109-118.  doi: 10.1007/BF00773449. [11] I. M. Singer and J. A. Thorpe, The curvature of 4-dimensional Einstein spaces, Global Analysis (papers in honour of K. Kodaira), Univ. of Tokyo Press 1969,355–365. [12] F. Tricerri and L. Vanhecke, Curvature tensors on almost hermitian manifolds, Trans. Amer. Math. Soc., 267 (1981), 365-397.  doi: 10.1090/S0002-9947-1981-0626479-0. [13] I. Vaisman, Symplectic curvature tensors, Monats. Math., 100 (1985), 299-327.  doi: 10.1007/BF01339231. [14] I. Vaisman, Variations on the theme of Twistor Spaces, Balkan J. Geom. Appl., 3 (1998), 135-156.

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