In mathematics, a web permits an intrinsic characterization in terms of Riemannian geometry of the additive separation of variables in the Hamilton–Jacobi equation.
Formal definition.
An orthogonal web on a Riemannian manifold "(M,g)" is a set formula_1 of "n" pairwise transversal and orthogonal foliations of connected submanifolds 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 formula_2 of "n" pairwise transversal and orthogonal foliations of connected submanifolds 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 foliation). 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 Blaschke in the 1930s. He extended the same group-theoretic approach to web geometry.
Classical definition.
Let formula_3 be a differentiable manifold of dimension "N=nr". A "d"-"web" "W(d,n,r)" of "codimension" "r" in an open set formula_4 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.
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