Reeb foliation
Encyclopedia
In mathematics
, the Reeb foliation is a particular foliation
of the 3-sphere
, introduced by the French mathematician Georges Reeb
(1920–1992).
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 the dividing torus surface forms one more leaf.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Reeb foliation is a particular 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....
of the 3-sphere
3-sphere
In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. It consists of the set of points equidistant from a fixed central point in 4-dimensional Euclidean space...
, introduced by the French mathematician Georges Reeb
Georges Reeb
Georges Henri Reeb was a French mathematician. He worked in differential topology, differential geometry, differential equations, topological dynamical systems theory and non-standard analysis....
(1920–1992).
It is based on dividing the sphere into two solid tori, along a 2-torus
Torus
In geometry, a torus is a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle...
: see Clifford torus
Clifford torus
In geometric topology, the Clifford torus is a special kind of torus sitting inside R4. Alternatively, it can be seen as a torus sitting inside C2 since C2 is topologically the same space as R4...
. Each of the solid tori is then foliated internally, in codimension
Codimension
In mathematics, codimension is a basic geometric idea that applies to subspaces in vector spaces, and also to submanifolds in manifolds, and suitable subsets of algebraic varieties.The dual concept is relative dimension.-Definition:...
1, and the dividing torus surface forms one more leaf.