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 on a Riemannian manifold of dimension n is a set 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. 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 be a differentiable manifold of dimension N=nr. A d-web W codimension r in an open set is a set of d foliations of codimension r which are in general position.
In the notation W 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.