On twistor almost complex structures

Michel Cahen; Simone Gutt; John Rawnsley

doi:10.3934/jgm.2021006

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September 2021, 13(3): 313-331. doi: 10.3934/jgm.2021006

On twistor almost complex structures

Michel Cahen ^1, , Simone Gutt ^1,, and John Rawnsley ^2,

1.	Membres de l'Académie Royale de Belgique, Université Libre de Bruxelles, Département de Mathématique, Campus Plaine, CP 218, boulevard du triomphe, BE-1050 Bruxelles, Belgium
2.	Mathematics Institute, Zeeman Building, University of Warwick, Coventry CV4 7AL, United Kingdom

^* Corresponding author: Simone Gutt

Dedicated to our friend Kirill Mackenzie

Received November 2020 Published September 2021 Early access April 2021

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.

Keywords: Riemannian geometry, symplectic geometry, almost complex structures, twistor spaces, Nijenhuis tensor.

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

Citation: Michel Cahen, Simone Gutt, John Rawnsley. On twistor almost complex structures. Journal of Geometric Mechanics, 2021, 13 (3) : 313-331. doi: 10.3934/jgm.2021006

References:

[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.

show all references

References:

[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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