web (differential geometry)

An orthogonal web on a Riemannian manifold (M,g) is a set $\backslash mathcal\; S\; =\; (\backslash mathcal\; S^1,\backslash dots,\backslash mathcal\; S^n)$ 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{{cite journal|authorlink=Gregorio Ricci-Curbastro|title=Dei sistemi di congruenze ortogonali in una varietà qualunque|author=G. Ricci-Curbastro|journal=Mem. Acc. Lincei |volume=2|year=1896|pages=276–322|issue=5}} $\backslash mathcal\; C\; =\; (\backslash mathcal\; C^1,\backslash dots,\backslash mathcal\; C^n)$ of n pairwise transversal and orthogonal foliations of connected submanifolds of dimension 1.

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.

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

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 $D\backslash subset\; X^\{nr\}$ 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.

• Foliation
• Parallelization (mathematics)

{{Reflist}}

• {{cite book | last = Sharpe | first = R. W. | title = Differential Geometry: Cartan's Generalization of Klein's Erlangen Program | publisher = Springer | location = New York | year=1997 | isbn=0-387-94732-9}}
• {{cite book |last1= Dillen |first1= F.J.E.| last2 = Verstraelen | first2 = L.C.A. | title = Handbook of Differential Geometry | publisher = North-Holland | location = Amsterdam | year=2000 |volume=1 | isbn=0-444-82240-2}}

Category:Differential geometry

Category:Manifolds

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