We define flag structures on a real three manifold M as the choice of two complex lines on the complexified tangent space at each point of M . We suppose that the plane field defined by the complex lines is a contact plane and construct an adapted connection on an appropriate principal bundle. This includes path geometries and CR structures as special cases. We prove that the null curvature models are given by totally real submanifolds in the flag space $$\mathbf{SL}(3,{{\mathbb {C}}})/B$$ SL ( 3 , C ) / B , where B is the subgroup of upper triangular matrices. We also define a global invariant which is analogous to the Chern–Simons secondary class invariant for three manifolds with a Riemannian structure and to the Burns–Epstein invariant in the case of CR structures. It turns out to be constant on homotopy classes of totally real immersions in flag space.

中文翻译：

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真实三流形上的标志结构

我们将实三流形 M 上的标志结构定义为在 M 的每个点的复切空间上选择两条复线。我们假设由复线定义的平面场是一个接触平面，并在适当的主丛上构造一个适配的连接。这包括路径几何和 CR 结构作为特殊情况。我们证明了零曲率模型由标志空间 $$\mathbf{SL}(3,{{\mathbb {C}}})/B$$ SL ( 3 , C ) / B 中的全实子流形给出，其中 B 是上三角矩阵的子群。我们还定义了一个全局不变量，它类似于具有黎曼结构的三个流形的 Chern-Simons 二级不变量和 CR 结构的 Burns-Epstein 不变量。