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Symplectic $ {\mathbb Z}_2^n $-manifolds
On twistor almost complex structures
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 |
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.
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. |
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M. Berger,
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A. L. Besse, Einstein Manifolds, Springer-Verlag, Berlin–Heidelberg, 1987.
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M. Cahen, M. Gérard, S. Gutt and M. Hayyani, Distributions associated to almost complex structures on symplectic manifolds, preprint, arXiv: 2002.02335. |
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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.
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[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.
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[7] |
J. Fine and D. Panov,
Hyperbolic geometry and non-Kähler manifolds with trivial canonical bundle, Geometry and Topology, 14 (2010), 1723-1763.
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[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.
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[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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