Reeb

Reeb is most commonly associated with Georges Reeb, a French mathematician whose work in differential topology and foliation theory gave rise to several foundational concepts that bear his name. The term Reeb appears in multiple mathematical constructions, including Reeb graphs, Reeb vector fields, Reeb foliations, and Reeb components, as well as in stability results related to foliations.

A Reeb graph is a construction in topology derived from a smooth function on a manifold. Given a function f: M → R, the Reeb graph is formed by collapsing each connected component of a level set f−1(c) to a single point. The resulting quotient space is a graph that encodes how level sets merge and split as the function value changes. Reeb graphs are used in Morse theory, shape analysis, and data visualization.

In contact geometry, the Reeb vector field is defined for a contact form α on an odd-dimensional manifold. The Reeb vector field R satisfies α(R) = 1 and iR dα = 0, and it generates the Reeb flow. This flow is central to the study of contact dynamics and has applications in Hamiltonian systems and symplectic topology.

The Reeb foliation is a classic example of a codimension-1 foliation of the 3-sphere S^3 introduced by Reeb. It features a Reeb component, a solid torus region where leaves spiral toward the boundary torus. Reeb components illustrate essential phenomena in foliation theory, such as the existence of nontrivial foliations on simple manifolds.

Reeb stability refers to Reeb’s results on foliations, notably a stability theorem describing when a compact leaf with finite holonomy has a neighborhood that is foliated as a product. This theorem provides fundamental insight into the local structure of foliations.

In summary, Reeb denotes a family of influential concepts in geometry and topology named after Georges Reeb, reflecting his impact on differential topology, foliation theory, and contact geometry.