标题： 双黎曼格拉斯曼流形的几何学 显示英文标题

标题： Geometry of bi-Lagrangian Grassmannian

摘要： 本文探讨了实或复向量空间上$2$-形式束的双拉格朗日格拉斯曼流形的结构。 我们将分析简化为若尔当-克罗内克标准形式仅由相同特征值的若尔当块组成的束。 我们证明这等价于研究在幂零自伴算子下不变的拉格朗日子空间。 我们计算了双拉格朗日格拉斯曼流形的维数，并描述了其在自同构群作用下的开轨道。 我们完全描述了以下三种情况下的自同构轨道：一个若尔当块的情况，相同若尔当块的和的情况以及两个不同若尔当块的和的情况。 显示英文摘要

摘要： This paper explores the structure of bi-Lagrangian Grassmanians for pencils of $2$-forms on real or complex vector spaces. We reduce the analysis to the pencils whose Jordan-Kronecker Canonical Form consists of Jordan blocks with the same eigenvalue. We demonstrate that this is equivalent to studying Lagrangian subspaces invariant under a nilpotent self-adjoint operator. We calculate the dimension of bi-Lagrangian Grassmanians and describe their open orbit under the automorphism group. We completely describe the automorphism orbits in the following three cases: for one Jordan block, for sums of equal Jordan blocks and for a sum of two distinct Jordan blocks.