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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.  Google Scholar

[2]

M. Berger, Sur quelques variétés riemaniennes suffisamment pincées, Bulletin de la S.M.F., 88 (1960), 57-71.   Google Scholar

[3]

A. L. Besse, Einstein Manifolds, Springer-Verlag, Berlin–Heidelberg, 1987. doi: 10.1007/978-3-540-74311-8.  Google Scholar

[4]

M. Cahen, M. Gérard, S. Gutt and M. Hayyani, Distributions associated to almost complex structures on symplectic manifolds, preprint, arXiv: 2002.02335. Google Scholar

[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.   Google Scholar

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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.   Google Scholar

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J. Fine and D. Panov, Hyperbolic geometry and non-Kähler manifolds with trivial canonical bundle, Geometry and Topology, 14 (2010), 1723-1763.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

A. G. Reznikov, Symplectic twistor spaces, Annals of Global Analysis and Geometry, 11 (1993), 109-118.  doi: 10.1007/BF00773449.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

I. Vaisman, Symplectic curvature tensors, Monats. Math., 100 (1985), 299-327.  doi: 10.1007/BF01339231.  Google Scholar

[14]

I. Vaisman, Variations on the theme of Twistor Spaces, Balkan J. Geom. Appl., 3 (1998), 135-156.   Google Scholar

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.  Google Scholar

[2]

M. Berger, Sur quelques variétés riemaniennes suffisamment pincées, Bulletin de la S.M.F., 88 (1960), 57-71.   Google Scholar

[3]

A. L. Besse, Einstein Manifolds, Springer-Verlag, Berlin–Heidelberg, 1987. doi: 10.1007/978-3-540-74311-8.  Google Scholar

[4]

M. Cahen, M. Gérard, S. Gutt and M. Hayyani, Distributions associated to almost complex structures on symplectic manifolds, preprint, arXiv: 2002.02335. Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[7]

J. Fine and D. Panov, Hyperbolic geometry and non-Kähler manifolds with trivial canonical bundle, Geometry and Topology, 14 (2010), 1723-1763.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

A. G. Reznikov, Symplectic twistor spaces, Annals of Global Analysis and Geometry, 11 (1993), 109-118.  doi: 10.1007/BF00773449.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

I. Vaisman, Symplectic curvature tensors, Monats. Math., 100 (1985), 299-327.  doi: 10.1007/BF01339231.  Google Scholar

[14]

I. Vaisman, Variations on the theme of Twistor Spaces, Balkan J. Geom. Appl., 3 (1998), 135-156.   Google Scholar

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