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In mathematics, the Reeb vector field, named after the French mathematician Georges Reeb, is a notion that appears in various domains of contact geometry including: in a contact manifold , given a contact 1-form α {\displaystyle \alpha } , the Reeb vector field satisfies R ∈ k e r d α , α ( R ) = 1 {\displaystyle R\in \mathrm {ker} \ d ...
A Reeb graph [1] (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. [2] According to [ 3 ] a similar concept was introduced by G.M. Adelson-Velskii and A.S. Kronrod and applied to analysis of Hilbert's thirteenth problem . [ 4 ]
Under certain conditions the Reeb local stability theorem may replace the Poincaré–Bendixson theorem in higher dimensions. [2] This is the case of codimension one, singular foliations ( M n , F ) {\displaystyle (M^{n},F)} , with n ≥ 3 {\displaystyle n\geq 3} , and some center-type singularity in S i n g ( F ) {\displaystyle Sing(F)} .
In 1978, Edward Wagneur generalized the Reeb sphere theorem to Morse foliations with saddles. He showed that the number of centers cannot be too much as compared with the number of saddles, notably, c ≤ s + 2 {\displaystyle c\leq s+2} .
In mathematics, the Reeb foliation is a particular foliation of the 3-sphere, introduced by the French mathematician Georges Reeb (1920–1993). It is based on dividing the sphere into two solid tori , along a 2- torus : see Clifford torus .
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Georges Henri Reeb (12 November 1920 – 6 November 1993) was a French mathematician. He worked in differential topology , differential geometry , differential equations , topological dynamical systems theory and non-standard analysis .
This is a very general concept, of which the Reeb graph [31] and merge trees are special cases. This is not quite the original definition. [ 29 ] Carlsson et al. choose Z {\displaystyle Z} to be R {\displaystyle \mathbb {R} } or R 2 {\displaystyle \mathbb {R} ^{2}} , and cover it with open sets such that at most two intersect. [ 3 ]