Nijenhuis 几何 III：gl-正则 Nijenhuis 算子
Nijnehuis Geometry III: gl-regular Nijenhuis operators
AI总结：
本文研究几何重数为一的 gl-正则 Nijenhuis 算子，建立其局部伴随形式与二维奇点标准形，并给出闭曲面上存在性的拓扑限制。
摘要：
我们研究 Nijenhuis 算子，即 Nijenhuis 挠率为零的 (1,1)-张量，并附加假设它们是 gl-正则的，也就是说每个特征值的几何重数均为一。我们证明存在一个坐标系，使该算子在其中取第一或第二伴随形式，并给出此类算子的局部描述。我们将这一局部描述应用于奇点研究。特别地，我们得到了它们在二维情形下的标准形，并揭示了闭曲面上存在 gl-正则 Nijenhuis 算子的拓扑限制。本文是 arXiv:1903.04603 与 arXiv:1903.06411 所提出研究纲领中的重要一步。
英文摘要：
We study Nijenhuis operators, that is, (1,1)-tensors with vanishing Nijenhuis torsion under the additional assumption that they are gl-regular, i.e., every eigenvalue has geometric multiplicity one. We prove the existence of a coordinate system in which the operator takes first or second companion form, and give a local describtion of such operators. We apply this local description to study singular points. In particular, we obtain their normal forms in dimension two and discover topological restrictions for the existence of gl-regular Nijenhuis operators on closed surfaces. This paper is an important step in the research programme suggested in arXiv:1903.04603 and arXiv:1903.06411.
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