Georges Reeb
Encyclopedia
Georges Henri Reeb was a French
mathematician
. He worked in differential topology
, differential geometry, differential equations, topological dynamical systems theory
and non-standard analysis
.
In 1943 he received his PhD from University of Strasbourg
with the dissertation Propriétés topologiques des variétés feuilletées. His adviser was Charles Ehresmann
.
In 1954, he was at the Institute for Advanced Study
.
In 1965 Reeb, Jean Leray
and Pierre Lelong founded a series of encounters between theoretical physicists and mathematicians in Strasbourg (Rencontres entre Mathématiciens et Physiciens Théoriciens).
He was a professor in Grenoble
(Université Joseph Fourier) and Strasbourg
(Université Louis Pasteur) where he directed the Institut de Recherche Mathématique Avancée (The Institute of Mathematics of the University of Strasbourg
) between 1967 and 1972, which he founded with Jean Frenkel in 1966.
Reeb is the founder of the topological theory of foliation
s (Feuilletées, Blätterungen), manifolds with a special local product structure.
He invented what is now called the Reeb foliation
, a foliation of the 3-sphere, all the leaves of which are diffeomorphic to R2, except one, which is a (compact!) 2-torus.
Reeb sphere theorem
says that a compact manifold with a function with exactly two critical points is homeomorphic to the sphere. This is used to prove that the Milnor spheres, although not diffeomorphic, are homeomorphic to the sphere S7, a result that came in 1956.
Reeb received an honorary doctorate from the Albert-Ludwigs-Universität Freiburg.
France
The French Republic , The French Republic , The French Republic , (commonly known as France , is a unitary semi-presidential republic in Western Europe with several overseas territories and islands located on other continents and in the Indian, Pacific, and Atlantic oceans. Metropolitan France...
mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
. He worked in differential topology
Differential topology
In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. It is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds.- Description :...
, differential geometry, differential equations, topological dynamical systems theory
Dynamical systems theory
Dynamical systems theory is an area of applied mathematics used to describe the behavior of complex dynamical systems, usually by employing differential equations or difference equations. When differential equations are employed, the theory is called continuous dynamical systems. When difference...
and non-standard analysis
Non-standard analysis
Non-standard analysis is a branch of mathematics that formulates analysis using a rigorous notion of an infinitesimal number.Non-standard analysis was introduced in the early 1960s by the mathematician Abraham Robinson. He wrote:...
.
In 1943 he received his PhD from University of Strasbourg
University of Strasbourg
The University of Strasbourg in Strasbourg, Alsace, France, is the largest university in France, with about 43,000 students and over 4,000 researchers....
with the dissertation Propriétés topologiques des variétés feuilletées. His adviser was Charles Ehresmann
Charles Ehresmann
Charles Ehresmann was a French mathematician who worked on differential topology and category theory. He is known for work on the topology of Lie groups, the jet concept , and his seminar on category theory.He attended the École Normale Supérieure in Paris before performing one year of military...
.
In 1954, he was at the Institute for Advanced Study
Institute for Advanced Study
The Institute for Advanced Study, located in Princeton, New Jersey, United States, is an independent postgraduate center for theoretical research and intellectual inquiry. It was founded in 1930 by Abraham Flexner...
.
In 1965 Reeb, Jean Leray
Jean Leray
Jean Leray was a French mathematician, who worked on both partial differential equations and algebraic topology....
and Pierre Lelong founded a series of encounters between theoretical physicists and mathematicians in Strasbourg (Rencontres entre Mathématiciens et Physiciens Théoriciens).
He was a professor in Grenoble
Grenoble
Grenoble is a city in southeastern France, at the foot of the French Alps where the river Drac joins the Isère. Located in the Rhône-Alpes region, Grenoble is the capital of the department of Isère...
(Université Joseph Fourier) and Strasbourg
Strasbourg
Strasbourg is the capital and principal city of the Alsace region in eastern France and is the official seat of the European Parliament. Located close to the border with Germany, it is the capital of the Bas-Rhin département. The city and the region of Alsace are historically German-speaking,...
(Université Louis Pasteur) where he directed the Institut de Recherche Mathématique Avancée (The Institute of Mathematics of the University of Strasbourg
University of Strasbourg
The University of Strasbourg in Strasbourg, Alsace, France, is the largest university in France, with about 43,000 students and over 4,000 researchers....
) between 1967 and 1972, which he founded with Jean Frenkel in 1966.
Reeb is the founder of the topological theory of foliation
Foliation
In mathematics, a foliation is a geometric device used to study manifolds, consisting of an integrable subbundle of the tangent bundle. A foliation looks locally like a decomposition of the manifold as a union of parallel submanifolds of smaller dimension....
s (Feuilletées, Blätterungen), manifolds with a special local product structure.
He invented what is now called the Reeb foliation
Reeb foliation
In mathematics, the Reeb foliation is a particular foliation of the 3-sphere, introduced by the French mathematician Georges Reeb .It is based on dividing the sphere into two solid tori, along a 2-torus: see Clifford torus. Each of the solid tori is then foliated internally, in codimension 1, and...
, a foliation of the 3-sphere, all the leaves of which are diffeomorphic to R2, except one, which is a (compact!) 2-torus.
Reeb sphere theorem
Reeb sphere theorem
In mathematics, Reeb sphere theorem, named after Georges Reeb, states that-Morse foliation:A singularity of a foliation F is of Morse type if in its small neighborhood all leaves of the foliation are levels of a Morse function, being the singularity a critical point of the function...
says that a compact manifold with a function with exactly two critical points is homeomorphic to the sphere. This is used to prove that the Milnor spheres, although not diffeomorphic, are homeomorphic to the sphere S7, a result that came in 1956.
Reeb received an honorary doctorate from the Albert-Ludwigs-Universität Freiburg.
See also
- Reeb graphReeb graphIn Morse theory, a branch of mathematics, a Reeb graph of a scalar function describes the connectivity of its level sets.Reeb graphs are named after Georges Reeb....
- Reeb vector fieldReeb vector fieldIn mathematics, the Reeb vector field, named after the French mathematician Georges Reeb, is a notion that appears in various domains of contact geometry including:...
- Contact geometryContact geometryIn mathematics, contact geometry is the study of a geometric structure on smooth manifolds given by a hyperplane distribution in the tangent bundle and specified by a one-form, both of which satisfy a 'maximum non-degeneracy' condition called 'complete non-integrability'...
- Séminaire Nicolas BourbakiSéminaire Nicolas BourbakiThe Séminaire Nicolas Bourbaki is a series of seminars that has been held in Paris since 1948. It is one of the major institutions of contemporary mathematics, and a barometer of mathematical achievement, fashion, and reputation...
- Séminaire Nicolas Bourbaki (1950–1959)
- Reeb sphere theoremReeb sphere theoremIn mathematics, Reeb sphere theorem, named after Georges Reeb, states that-Morse foliation:A singularity of a foliation F is of Morse type if in its small neighborhood all leaves of the foliation are levels of a Morse function, being the singularity a critical point of the function...
- Reeb foliationReeb foliationIn mathematics, the Reeb foliation is a particular foliation of the 3-sphere, introduced by the French mathematician Georges Reeb .It is based on dividing the sphere into two solid tori, along a 2-torus: see Clifford torus. Each of the solid tori is then foliated internally, in codimension 1, and...
- Reeb stability theoremReeb stability theoremIn mathematics, Reeb stability theorem, named after Georges Reeb, asserts that if one leaf of a codimension-one foliation is closed and has finite fundamental group, then all the leaves are closed and have finite fundamental group....