We study Morse-Bott functions with two critical values (equivalently, non-constant without saddles) on closed surfaces. We show that only four surfaces admit such functions (though in higher dimensions, we construct many such manifolds, e.g. as fiber bundles over already constructed manifolds with the same property). We study properties of such functions. Namely, their Reeb graphs are path or cycle graphs; any path graph, and any cycle graph with an even number of vertices, is isomorphic to the Reeb graph of such a function. They have a specific number of center singularities and singular circles with nonorientable normal bundle, and an unlimited number (with some conditions) of singular circles with orientable normal bundle. They can, or cannot, be chosen as the height function associated with an immersion of the surface in the three-dimensional space, depending on the surface and the Reeb graph. In addition, for an arbitrary Morse-Bott function on a closed surface, we show that the Euler characteristic of the surface is determined by the isolated singularities and does not depend on the singular circles.

我们研究在闭合表面上具有两个临界值（等效地，没有鞍形的非常数）的Morse-Bott函数。我们证明只有四个表面具有这种功能（尽管在更大的尺寸上，我们构造了许多这样的歧管，例如，在已经构造的具有相同特性的歧管上形成纤维束）。我们研究这种功能的性质。也就是说，它们的Reeb图是路径图或循环图；任何路径图和具有偶数个顶点的任何循环图都与该函数的Reeb图同构。它们具有特定数量的中心奇点和具有不可定向法向束的奇异圆，以及无限数量（在某些条件下）具有可定向法向束的奇异圆。他们可以或不能 取决于曲面和Reeb图，选择“ R”作为与曲面在三维空间中的浸入相关的高度函数。此外，对于封闭表面上的任意Morse-Bott函数，我们证明了该表面的欧拉特性由孤立的奇点确定，并且不依赖于奇数圆。