Reeb spaces of smooth functions associated to globally similar graphs of smooth functions

Previously, we have investigated a natural smooth map onto the region surrounded by the graphs of two smooth real-valued functions in the plane converging to a same value or diverges to $+\infty$ or $-\infty$ simultaneously, at each infinity, and topological properties and combinatorial ones of its composition with the canonical projection. Here, we consider smooth functions with congruent or globally similar graphs instead. Here, the Reeb space of a smooth function on a manifold with no boundary is fundamental and important. This is the naturally topologized quotient space of the manifold, consisting of all connected components (contours) of the function and is a graph under a certain nice situation. Related studies also related to the present study were started due to interest of the author in theory of Reeb spaces of non-proper functions. For proper functions, in 2020s related studies have developed mainly due to Gelbukh and Saeki.

💡 Research Summary

The paper investigates the Reeb spaces of smooth real‑valued functions that are defined by the region sandwiched between the graphs of two smooth functions c₁ and c₂ whose graphs are globally similar (congruent up to a linear transformation, rotation, or scaling). Building on the author’s earlier work where the two graphs were required to converge to the same limit at each infinity, the present study relaxes this condition and treats the case where the graphs are globally alike.

The central construction proceeds as follows. For any pair of smooth functions c₁, c₂ : ℝ → ℝ satisfying c₁(x) < c₂(x) for all x, the open strip
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