闭流形上高度函数及其一阶导数的基础例子
Height functions on products of spheres and associated Reeb digraphs and level sets
AI总结：
该研究聚焦闭流形上高度函数及其一阶导数，系统构造了相关高度函数映射，研究其全局行为与拓扑组合性质，关注Morse函数等广义类函数一阶导数的新挑战。
AI中文摘要：
高度函数在数学中是基础且重要的，尤其在几何学中，更明确地说，是可微映射的奇点理论，以及在微分拓扑和微分几何中的应用。包括作者在内的我们对构造到欧几里得空间区域的显式映射，以及将这些映射与典范投影复合得到的函数感兴趣。作者一直关注并成功系统地构造了以高度函数为函数的这类映射，这在可微映射的奇点理论中很重要。我们研究这类函数一阶导数的全局行为，作者也关注这类函数的拓扑性质和组合性质。对于所谓的Morse函数及广义类函数，这些性质仍主要由Gelbukh和Michalak积极研究，其一阶导数是一类新的挑战。
英文摘要：
Height functions are fundamental and important objects and tools in mathematics, especially in geometry such as differential topology and differential geometry and some related singularity theory of differentiable maps. Our interest, especially interest of the author, lies in obtaining explicit lists of such functions. Recently, as information on so-called higher degrees, he is also interested in their naturally defined 1st derivatives. We consider natural maps on products of spheres which are variants of specific cases of so-called moment maps on toric symplectic manifolds. We also generalize cases of the canonical projections of the unit spheres. This is a further result on related previous study of the author. We use Reeb graphs, graphs being natural quotient spaces of manifolds of the domains of nice functions such as Morse-Bott functions, and consisting of connected components of level sets. They are fundamental tools and objects since the last century.
