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.^[1][2]

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 ${\displaystyle {\mathcal {S}}=({\mathcal {S}}^{1},\dots ,{\mathcal {S}}^{n})}$ of n pairwise transversal and orthogonal foliations of connected submanifolds of codimension 1.^[3] 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 ${\displaystyle M=X^{nr}}$ be a differentiable manifold of dimension N=nr. A d-web W(d,n,r) codimension r in an open set ${\displaystyle D\subset X^{nr}}$ 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)