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On twistor almost complex structures

  • * Corresponding author: Simone Gutt

    * Corresponding author: Simone Gutt 

Dedicated to our friend Kirill Mackenzie

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  • 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.


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