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 ...

, a web permits an intrinsic characterization in terms of

Riemannian geometry Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a ''Riemannian metric'', i.e. with an inner product on the tangent space at each point that varies smoothly from point to poin ...

of the additive separation of variables in the

Hamilton–Jacobi equation In physics, the Hamilton–Jacobi equation, named after William Rowan Hamilton and Carl Gustav Jacob Jacobi, is an alternative formulation of classical mechanics, equivalent to other formulations such as Newton's laws of motion, Lagrangian mechan ...

Formal definition

orthogonal In mathematics, orthogonality is the generalization of the geometric notion of ''perpendicularity''. By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in ...

web on a

Riemannian manifold In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real manifold, real, smooth manifold ''M'' equipped with a positive-definite Inner product space, inner product ...

''(M,g)'' is a set

\mathcal S = (\mathcal S^1,\dots,\mathcal S^n)

of ''n'' pairwise transversal and orthogonal

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 ...

s of connected

submanifold In mathematics, a submanifold of a manifold ''M'' is a subset ''S'' which itself has the structure of a manifold, and for which the inclusion map satisfies certain properties. There are different types of submanifolds depending on exactly which p ...

s of codimension ''1'' and where ''n'' denotes the

dimension In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...

of ''M''. Note that two submanifolds of codimension ''1'' are orthogonal if their normal vectors are orthogonal and in a nondefinite metric orthogonality does not imply transversality.

Alternative definition

Given a smooth manifold of dimension ''n'', an

web (also called orthogonal grid or Ricci’s grid) on a

''(M,g)'' is a set

\mathcal C = (\mathcal C^1,\dots,\mathcal C^n)

of ''n'' pairwise transversal and orthogonal

s of connected

s of dimension ''1''.

Remark

Since vector fields can be visualized as stream-lines of a stationary flow or as Faraday’s lines of force, a non-vanishing vector field in space generates a space-filling system of lines through each point, known to mathematicians as a congruence (i.e., a local

Ricci Ricci () is an Italian surname, derived from the adjective "riccio", meaning curly. Notable Riccis Arts and entertainment * Antonio Ricci (painter) (c.1565–c.1635), Spanish Baroque painter of Italian origin * Christina Ricci (born 1980), Ameri ...

’s vision filled Riemann’s ''n''-dimensional manifold with ''n'' congruences orthogonal to each other, i.e., a local orthogonal grid.

Differential geometry of webs

A systematic study of webs was started by Blaschke in the 1930s. He extended the same group-theoretic approach to web geometry.

Classical definition

Let

M=X^

be a differentiable manifold of dimension ''N=nr''. A ''d''-''web'' ''W(d,n,r)'' of ''codimension'' ''r'' in an open set

D\subset X^

is a set of ''d'' foliations of codimension ''r'' which are in general position. In the notation ''W(d,n,r)'' the number ''d'' is the number of foliations forming a web, ''r'' is the web codimension, and ''n'' is the ratio of the dimension ''nr'' of the manifold ''M'' and the web codimension. Of course, one may define a ''d''-''web'' of codimension ''r'' without having ''r'' as a divisor of the dimension of the ambient manifold.

Notes

References

* * Differential geometry Manifolds {{differential-geometry-stub