mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, a web permits an intrinsic characterization in terms of

Riemannian geometry Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, defined as manifold, smooth manifolds with a ''Riemannian metric'' (an inner product on the tangent space at each point that varies smooth function, smo ...

of the additive separation of variables in the

Hamilton–Jacobi equation In physics, the Hamilton–Jacobi equation, named after William Rowan Hamilton and Carl Gustav Jacob Jacobi, is an alternative formulation of classical mechanics, equivalent to other formulations such as Newton's laws of motion, Lagrangian mecha ...

Formal definition

orthogonal In mathematics, orthogonality (mathematics), orthogonality is the generalization of the geometric notion of ''perpendicularity''. Although many authors use the two terms ''perpendicular'' and ''orthogonal'' interchangeably, the term ''perpendic ...

web (also called an orthogonal grid or Ricci grid) on a

Riemannian manifold In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the N-sphere, n-sphere, hyperbolic space, and smooth surf ...

''(M,g)'' of dimension ''n'' is a set

\mathcal S = (\mathcal S^1,\dots,\mathcal S^n)

of ''n'' pairwise transversal and orthogonal

foliation In mathematics (differential geometry), a foliation is an equivalence relation on an topological manifold, ''n''-manifold, the equivalence classes being connected, injective function, injectively immersed submanifolds, all of the same dimension ...

s of connected

submanifold In mathematics, a submanifold of a manifold M is a subset S which itself has the structure of a manifold, and for which the inclusion map S \rightarrow M satisfies certain properties. There are different types of submanifolds depending on exactly ...

s of codimension ''1''. Note that two submanifolds of codimension ''1'' are orthogonal iff their normal vectors are orthogonal, and that in the case of a nondefinite metric, orthogonality does not imply transversality.

Remark

Since

vector field In vector calculus and physics, a vector field is an assignment of a vector to each point in a space, most commonly Euclidean space \mathbb^n. A vector field on a plane can be visualized as a collection of arrows with given magnitudes and dire ...

s 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

). Ricci’s idea was to fill an ''n''-dimensional Riemannian 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 Blaschke in the 1930s. He extended the same group-theoretic approach to web geometry.

Classical definition

Let

M=X^

be a differentiable manifold of dimension ''N=nr''. A ''d-web W(d,n,r) codimension r'' in an open set

D\subset X^

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.

Notes

References