Web (differential geometry)

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 (also called an orthogonal grid or Ricci grid) on a Riemannian manifold (M,g) of dimension n is a set $𝒮=({𝒮}^1,\dots ,{𝒮}^n)$ of n pairwise transversal and orthogonal foliations of connected submanifolds of codimension 1. Note that two submanifolds of codimension 1 are orthogonal iff their normal vectors are orthogonal, and that in the case of a nondefinite metric, orthogonality does not imply transversality.

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 idea was to fill an n-dimensional Riemannian 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 $M=X^{nr}$ be a differentiable manifold of dimension N=nr. A d-web W(d,n,r) codimension r in an open set $D\subset X^{nr}$ 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
• Parallelization (mathematics)