# Morse–Bott functions with two critical values on a surface

**Author:** Irina Gelbukh

Czechoslovak Mathematical Journal,
2020.

**Abstract:**
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 non-orientable 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.

**Keywords:** Morse–Bott function, height function, surface, critical values, Reeb graph

**PDF:** Morse–Bott functions with two critical values on a surface

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