Characterization of homogeneous spaces admitting flat geometric structures by means of invariants

我们研究了齐次空间(或李群)上的几个条件，以便从不变理论的观点看它们允许平坦的几何结构。在这项研究中，我们主要讨论射影结构和伪黎曼结构，在某些情况下，我们成功地得到了允许平坦几何结构的几个必要条件。我们还利用表示理论构造了允许平坦射影结构的新的齐次空间，并确定了每个三维李群上几乎平坦的伪黎曼结构的存在性或不存在性。

We study several conditions on homogeneous spaces (or Lie groups) in order that they admit flat geometric structures from the standpoint of invariant theory. In this research, we treat mainly projective and pseudo-Riemannian structures, and we succeed to obtain several necessary conditions to admit flat geometric structures for some cases. We also construct new homogeneous spaces admitting flat projective structures by using representation theory, and in addition, determine the existence or non-existence of almost flat pseudo-Riemannian structures on each 3-dimensional Lie group.