\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

On twistor almost complex structures

  • * Corresponding author: Simone Gutt

    * Corresponding author: Simone Gutt 

Dedicated to our friend Kirill Mackenzie

Abstract Full Text(HTML) Related Papers Cited by
  • 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:

    \begin{equation} \\ \end{equation}
  • 加载中
  • [1] M. F. AtiyahN. 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. 
  • 加载中
SHARE

Article Metrics

HTML views(721) PDF downloads(255) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return