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Fix CS1 deprecated date parameter errors

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In [[mathematics]], a '''web''' permits an intrinsic characterization in terms of [[Riemannian geometry]] of the additive separation of variables in the [[Hamilton–Jacobi equation]].<ref>{{cite journal|title=Intrinsic characterization of the variable separation in the Hamilton-Jacobi equation |author=S. Benenti|journal=J. Math. Phys. |volume=38|year=1997|pages=6578–6602|issue=12}}</ref><ref>{{cite journal | title= Eigenvalues of Killing Tensors and Separable Webs on
Riemannian and Pseudo-Riemannian Manifolds | last1 = Chanu | first1= Claudia | last2 = Rastelli|first2 = Giovanni | journal=SIGMA | volume=3 | year=2007 | pages=021, 21 pp }}</ref>

==Formal definition==
An [[orthogonal]] '''web''' on a [[Riemannian manifold]] ''(M,g)'' is a set <math>\mathcal S = (\mathcal S^1,\dots,\mathcal S^n)</math> of ''n'' pairwise [[transversal (geometry)|transversal]] and orthogonal [[foliation]]s of connected [[submanifold]]s of codimension ''1'' and where ''n'' denotes the [[dimension]] 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 [[orthogonal]] '''web''' (also called '''orthogonal grid''' or '''Ricci’s grid''') on a [[Riemannian manifold]] ''(M,g)'' is a set<ref>{{cite journal|authorlink=Gregorio Ricci-Curbastro|title=Dei sistemi di congruenze ortogonali in una varietà qualunque|author=G. Ricci-Curbastro|journal=Mem. Acc. Lincei |volume=2|year=1896|pages=276–322|issue=5}}</ref> <math>\mathcal C = (\mathcal C^1,\dots,\mathcal C^n)</math> of ''n'' pairwise [[transversal (geometry)|transversal]] and orthogonal [[foliation]]s of connected [[submanifold]]s of dimension ''1''.
===Remark===
Since [[vector field]]s 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 (manifolds)|congruence]] (i.e., a local [[foliation]]). [[Gregorio Ricci-Curbastro|Ricci]]’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 [[Wilhelm Blaschke|Blashke]] in the 1930s. He extended the same group-theoretic approach to web geometry.
===Classical definition===
Let <math>M=X^{nr}</math> be a differentiable manifold of dimension ''N=nr''. A ''d''-''web'' ''W(d,n,r)'' of ''codimension'' ''r'' in an open set <math>D\subset X^{nr}</math> 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.
==See also==
* [[Foliation]]

==Notes==
{{Reflist}}

==References==
*{{cite book | last = Sharpe | first = R. W. | title = Differential Geometry: Cartan's Generalization of Klein's Erlangen Program | publisher = Springer | location = New York | year=1997 | isbn=0-387-94732-9}}
*{{cite book |last1= Dillen |first1= F.J.E.| last2 = Verstraelen | first2 = L.C.A. | title = Handbook of Differential Geometry | publisher = North-Holland | location = Amsterdam | year=2000 |volume=Volume 1 | isbn=0-444-82240-2}}

[[Category:Differential geometry]]
[[Category:Manifolds]]

{{differential-geometry-stub}}

Multiscroll attractor - Revision history

en>Monkbot: Fix CS1 deprecated date parameter errors