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.

An orthogonal web on a Riemannian manifold (M,g) 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 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.

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${\displaystyle {\mathcal {C}}=({\mathcal {C}}^{1},\dots ,{\mathcal {C}}^{n})}$ of n pairwise transversal and orthogonal foliations of connected submanifolds of dimension 1.

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