Inmathematics, aweb permits an intrinsic characterization in terms ofRiemannian geometry of the additive separation of variables in theHamilton–Jacobi equation.^[1][2]

Anorthogonalweb on aRiemannian manifold(M,g) is a set${\displaystyle \mathcal{S}=({\mathcal{S}}^1,\dots ,{\mathcal{S}}^n)}$ ofn pairwisetransversal and orthogonalfoliations of connectedsubmanifolds of codimension1 and wheren denotes thedimension ofM.

Note that two submanifolds of codimension1 are orthogonal if their normal vectors are orthogonal and in a nondefinite metric orthogonality does not imply transversality.

Given a smooth manifold of dimensionn, anorthogonalweb (also calledorthogonal grid orRicci’s grid) on aRiemannian manifold(M,g) is a set^[3]${\displaystyle \mathcal{C}=({\mathcal{C}}^1,\dots ,{\mathcal{C}}^n)}$ ofn pairwisetransversal and orthogonalfoliations of connectedsubmanifolds of dimension1.

Sincevector 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 acongruence (i.e., a localfoliation).Ricci’s vision filled Riemann’sn-dimensional manifold withn congruences orthogonal to each other, i.e., a localorthogonal grid.

A systematic study of webs was started byBlaschke in the 1930s. He extended the same group-theoretic approach to web geometry.

Let${\displaystyle M=X^{nr}}$ be a differentiable manifold of dimensionN=nr. Ad-webW(d,n,r) ofcodimensionr in an open set${\displaystyle D\subset X^{nr}}$ is a set ofd foliations of codimensionr which are in general position.

In the notationW(d,n,r) the numberd is the number of foliations forming a web,r is the web codimension, andn is the ratio of the dimensionnr of the manifoldM and the web codimension. Of course, one may define ad-web of codimensionr without havingr as a divisor of the dimension of the ambient manifold.

• Foliation
• Parallelization (mathematics)

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

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