mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, the Reeb foliation is a particular

foliation In mathematics (differential geometry), a foliation is an equivalence relation on an ''n''-manifold, the equivalence classes being connected, injectively immersed submanifolds, all of the same dimension ''p'', modeled on the decomposition of ...

of the

3-sphere In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. It may be embedded in 4-dimensional Euclidean space as the set of points equidistant from a fixed central point. Analogous to how the boundary of a ball in three dimensi ...

, introduced by the French mathematician

Georges Reeb 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. Biography ...

(1920–1993). It is based on dividing the sphere into two solid tori, along a 2-

torus In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle. If the axis of revolution does not tou ...

: see

Clifford torus In geometric topology, the Clifford torus is the simplest and most symmetric flat embedding of the cartesian product of two circles ''S'' and ''S'' (in the same sense that the surface of a cylinder is "flat"). It is named after William Kingdo ...

. Each of the solid tori is then foliated internally, in

codimension In mathematics, codimension is a basic geometric idea that applies to subspaces in vector spaces, to submanifolds in manifolds, and suitable subsets of algebraic varieties. For affine and projective algebraic varieties, the codimension equals the ...

1, and the dividing torus surface forms one more leaf. By

Novikov's compact leaf theorem {{Short description, Result about foliation of compact 3-manifolds In mathematics, Novikov's compact leaf theorem, named after Sergei Novikov, states that : ''A codimension-one foliation of a compact 3-manifold whose universal covering space is n ...

, every smooth foliation of the 3-sphere includes a compact torus leaf, bounding a solid torus foliated in the same way.

Illustrations

References

External links