AI 中文总结
该研究针对一类固有浸没的一阶导数，探究其Reeb空间，补充了非固有情形外相关理论的新案例，完善了Morse函数理论中Reeb空间的相关研究。
AI 中文摘要
我们研究一类被表示为高度函数的固有浸没的（典范）一阶导数，固有映射指的是紧致集的原像始终紧致的映射。我们探究它们的Reeb空间，Reeb空间是由定义域流形上的等价关系定义的商空间，该等价关系将同一水平集同一连通分支中的两点视为等同。自20世纪Morse函数理论建立以来，Reeb空间就具有重要意义，在特定的驯服情形下，它们是0维或1维的，自然呈现为图状。2020年代，Gelbukh和Saeki针对某些固有光滑实值函数给出了相关结果；非固有情形下，除作者此前给出的个别情况外，构建相关显式理论十分困难，本研究聚焦于一个新的相关情形。
英文摘要
We study the (canonical) 1st derivatives of {\it proper} submersions represented as height functions and belonging to a certain class: a proper map means a map the preimage of a compact set by which is always compact. We investigate their {\it Reeb spaces}. They are the quotient spaces defined by the equivalence relations on the manifolds of the domains where we identify two points in a same connected component of a same level set of them. They have been important since the establishment of theory of Morse functions, in the 20th century, They are in certain tame situations $0$- or $1$-dimensional and graphs naturally. Related facts have been shown by Gelbukh and Saeki in the 2020s for certain proper smooth real-valued functions. In non-proper cases, even related explicit theory has been difficult, except some previously given case of the author. Our study is on a new related case.
Comments13 pages
