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 (M,g) is a set ${\mathcal {S}}=({\mathcal {S}}^{1},\dots ,{\mathcal {S}}^{n})$ 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 ${\mathcal {C}}=({\mathcal {C}}^{1},\dots ,{\mathcal {C}}^{n})$ 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 Blashke 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) of 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

References

• Sharpe, R. W. (1997). Differential Geometry: Cartan's Generalization of Klein's Erlangen Program. New York: Springer. ISBN 0-387-94732-9.
• Dillen, F.J.E.; Verstraelen, L.C.A. (2000). Handbook of Differential Geometry. Vol. Volume 1. Amsterdam: North-Holland. ISBN 0-444-82240-2. {{cite book}}: |volume= has extra text (help)