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.

中文翻译：

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关于扭曲几乎复杂的结构

在本文中，我们研究了在偶数维的扭曲空间 $ J(M, g) $ 上定义的两个自然的几乎复杂的结构 $ J^{\pm}_\nabla $ 的可积性与否的问题流形 $ M $ 与附加结构 $ g $ 和 $ \nabla $ a $ g $-connection。我们通过 $ N^{J^\pm_\nabla} $ 值的跨度维度来衡量它们的不可积性。我们也看看$J^{\pm}_\nabla $ 与定义在$J 上的自然封闭$2 $-form $\omega^{J(M,g,\nabla)}$ 的兼容性问题(M, g) $。对于 $ (M, g) $，我们考虑具有 Levi Civita 连接的伪黎曼流形（可定向或不可定向）或具有给定辛连接 $ \nabla $ 的辛流形。在所有情况下，$J(M,g)$ 都是与 $g$ 兼容的 $M$ 切空间上的一组复杂结构。在 $ M $ 是定向的情况下，我们要求复杂结构的方向是给定的。在辛情况下，复结构是正的。