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arXiv 2010.04780math.DGmath.SG
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关于扭振殆复结构

On Twistor Almost Complex Structures

Michel Cahen, Simone Gutt, John Rawnsley

备注：
21 pages; fixed some typos and re-laid some formulas for easier reading
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摘要：

在本文中，我们研究了定义在偶维流形$M$的扭量空间$J(M,g)$上的两个自然的殆复结构$J^{\pm}_\nabla$与附加结构$g$和$\nabla$ a $g$ -联络的可积性问题.我们也看看问题的兼容性$J^{\pm}_\nabla$与一个自然封闭的$2$ -形式$ω^{J(M,g,\nabla)}$定义的$J(M,g)$。对于$(M,g)$，我们考虑一个伪黎曼流形，定向或不定向，与Levi Civita联络或辛流形与给定的辛联络$\nabla$。在所有情况下，$J(M,g)$都是$M$的切空间上与$g$相容的复结构的丛，我们用$π\colon J(M,g) \longrightarrow M$表示丛投影。在$M$是定向的情况下，我们要求复杂结构的定向是给定的。在辛的情况下，复结构是正的。

英文摘要：

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 also look at the question of the compatibility of $J^{\pm}_\nabla$ with a natural closed $2$-form $ω^{J(M,g,\nabla)}$ 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 $g$ and we denote by $π\colon J(M,g) \longrightarrow M$ the bundle projection. 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.

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