Reeb graph

Summary

A Reeb graph (named after Georges Reeb by René Thom) is a mathematical object reflecting the evolution of the level sets of a real-valued function on a manifold. A similar concept was introduced by G.M. Adelson-Velskii and A.S. Kronrod and applied to analysis of Hilbert's thirteenth problem. Proposed by G. Reeb as a tool in Morse theory, Reeb graphs are the natural tool to study multivalued functional relationships between 2D scalar fields $\psi$, $\lambda$, and $\phi$ arising from the conditions $\nabla \psi = \lambda \nabla \phi$ and $\lambda \neq 0$, because these relationships are single-valued when restricted to a region associated with an individual edge of the Reeb graph. This general principle was first used to study neutral surfaces in oceanography.