Web (differential geometry)

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[3] \( \mathcal C = (\mathcal C^1,\dots,\mathcal C^n) \) 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

Notes

S. Benenti (1997). "Intrinsic characterization of the variable separation in the Hamilton-Jacobi equation". J. Math. Phys. 38 (12): 6578–6602. doi:10.1063/1.532226.
Chanu, Claudia; Rastelli, Giovanni (2007). "Eigenvalues of Killing Tensors and Separable Webs on Riemannian and Pseudo-Riemannian Manifolds". SIGMA 3: 021, 21 pp. doi:10.3842/sigma.2007.021.

G. Ricci-Curbastro (1896). "Dei sistemi di congruenze ortogonali in una varietà qualunque". Mem. Acc. Lincei 2 (5): 276–322.

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. Volume 1. Amsterdam: North-Holland. ISBN 0-444-82240-2.