View source for Net (in differential geometry)

A system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066340/n0663401.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066340/n0663402.png" /> families <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066340/n0663403.png" /> of sufficiently smooth curves in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066340/n0663404.png" /> of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066340/n0663405.png" />-dimensional differentiable manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066340/n0663406.png" /> such that: 1) through each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066340/n0663407.png" /> there passes exactly one curve of each family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066340/n0663408.png" />; and 2) the tangent vectors to these curves at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066340/n0663409.png" /> form a basis for the tangent space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066340/n06634010.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066340/n06634011.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066340/n06634012.png" />. The tangent vectors to the curves of one family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066340/n06634013.png" /> belong to a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066340/n06634014.png" />-dimensional distribution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066340/n06634015.png" /> defined in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066340/n06634016.png" />. The conditions for a family of curves to form a net in a certain neighbourhood of a point need not hold when the curves are extended. The curves of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066340/n06634017.png" /> are the integral curves of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066340/n06634018.png" />. A net <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066340/n06634019.png" /> is defined by specifying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066340/n06634020.png" /> one-dimensional distributions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066340/n06634021.png" /> such that the tangent space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066340/n06634022.png" /> at each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066340/n06634023.png" /> is the direct sum of the subspaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066340/n06634024.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066340/n06634025.png" />. A net <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066340/n06634026.png" /> defines in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066340/n06634027.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066340/n06634028.png" />-dimensional distributions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066340/n06634029.png" /> such that at each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066340/n06634030.png" /> the subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066340/n06634031.png" /> is the direct sum of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066340/n06634032.png" /> one-dimensional subspaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066340/n06634033.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066340/n06634034.png" />. The following types of nets are distinguished: holonomic nets, partially holonomic nets, for which certain of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066340/n06634035.png" /> are integrable and the remainder are not (such nets are distinguished according to the number of non-integrable distributions), and non-holonomic nets, for which all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066340/n06634036.png" /> are non-integrable.

If a distribution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066340/n06634037.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066340/n06634038.png" />, is integrable and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066340/n06634039.png" /> is an integral curve of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066340/n06634040.png" />, then through each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066340/n06634041.png" /> there passes an integral manifold of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066340/n06634042.png" /> that carries a net <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066340/n06634043.png" /> of curves belonging to the families <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066340/n06634044.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066340/n06634045.png" />.

A net <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066340/n06634046.png" /> can also be defined by one of the following means: a) by a system of vector fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066340/n06634047.png" />; b) by a system of differential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066340/n06634048.png" />-forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066340/n06634049.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066340/n06634050.png" />; or c) by the field of an affinor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066340/n06634051.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066340/n06634052.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066340/n06634053.png" /> is the identity affinor).

In the study of nets there are three basic problems: the intrinsic properties of nets, the exterior properties and an investigation of diffeomorphisms of nets.

The intrinsic properties of nets are induced by the structure of the manifold that carries the net. For example, a net <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066340/n06634054.png" /> in a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066340/n06634055.png" /> with an affine connection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066340/n06634056.png" /> is called geodesic if all its curves are geodesic. If a Riemannian manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066340/n06634057.png" /> with a torsion-free connection in which the metric tensor is covariantly constant carries an orthogonal [[Chebyshev net|Chebyshev net]] of the first kind, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066340/n06634058.png" /> is locally Euclidean. The connection of such nets with parallel transfer of vectors on a surface was established by L. Bianchi (1922). This connection is at the basis of A.P. Norden's definition of a Chebyshev net of the first kind in a space with an affine connection.

The exterior properties of a net are induced by the structure of the ambient space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066340/n06634059.png" />. For example, suppose that a net <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066340/n06634060.png" /> defined in a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066340/n06634061.png" /> on a smooth surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066340/n06634062.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066340/n06634063.png" />-dimensional projective space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066340/n06634064.png" /> is conjugate, that is, at each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066340/n06634065.png" /> the directions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066340/n06634066.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066340/n06634067.png" /> of the tangents to any two curves of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066340/n06634068.png" /> through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066340/n06634069.png" /> are conjugate (two directions are conjugate if each belongs to the characteristic of the tangent plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066340/n06634070.png" /> when it is displaced in the other direction). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066340/n06634071.png" /> is not contained in a projective space of dimension less than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066340/n06634072.png" />, then for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066340/n06634073.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066340/n06634074.png" /> carries an infinite set of conjugate nets; for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066340/n06634075.png" /> the surface carries, in general, a unique conjugate net, but there are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066340/n06634076.png" />-dimensional surfaces on which there are no conjugate nets; for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066340/n06634077.png" /> only <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066340/n06634078.png" />-dimensional surfaces of a special structure carry a conjugate net. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066340/n06634079.png" /> a conjugate net need not be holonomic (see [[#References|[3]]]). A special case of a holonomic conjugate net is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066340/n06634081.png" />-conjugate system: A net <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066340/n06634082.png" /> with the property that the tangents to the curves of each family taken along any curve of any other family form a developable surface. Conjugate systems exist in a projective space of any dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066340/n06634083.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066340/n06634084.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066340/n06634085.png" />. Surfaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066340/n06634086.png" /> that carry an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066340/n06634087.png" />-conjugate system in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066340/n06634088.png" />-dimensional projective space, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066340/n06634089.png" />, and for which at each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066340/n06634090.png" /> the osculating space (the space of second differentials of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066340/n06634091.png" />) has dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066340/n06634092.png" />, were first considered by E. Cartan [[#References|[4]]] under the name  "manifolds of special projective type"  (Cartan surfaces). The concept of the [[Laplace transformation (in geometry)|Laplace transformation (in geometry)]] was extended to such nets (see [[#References|[5]]], [[#References|[6]]]).

In the study of diffeomorphisms of nets, in terms of known properties of a net <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066340/n06634093.png" /> one describes properties of the net <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066340/n06634094.png" /> for a given diffeomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066340/n06634095.png" /> (for example, under a bending deformation or under a conformal mapping of a surface carrying a net), or one looks for a diffeomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066340/n06634096.png" /> that preserves certain properties of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066340/n06634097.png" />. For example, a net <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066340/n06634098.png" /> on a surface in Euclidean space is called a rhombic net (a conformal Chebyshev net) if it admits a conformal mapping onto a Chebyshev net. On every surface of revolution the [[Asymptotic net|asymptotic net]] is rhombic.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.P. Norden,   "Spaces with an affine connection" , Nauka , Moscow-Leningrad  (1976)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  Ya.S. Dubnov,   S.A. [S.A. Fuks] Fuchs,   "Sur quelques réseaux de l'espace analogues au réseau de Tchebychev"  ''Dokl. Akad. Nauk SSSR'' , '''28''' :  2  (1940)  pp. 102–105</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  V.T. Bazylev,   "Multidimensional nets and their transformations"  ''Itogi Nauk. Geom. 1963''  (1965)  pp. 138–164  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  E. Cartan,   "Sur les variétés de courbure constante d'un espace euclidien ou non euclidien"  ''Bull. Soc. Math. France'' , '''47'''  (1919)  pp. 125–160</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  S.S. Chern,   "Laplace transforms of a class of higher dimensional varieties in a projective space of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066340/n06634099.png" /> dimensions"  ''Proc. Nat. Acad. Sci. USA'' , '''30'''  (1944)  pp. 95–97</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  R.V. Smirnov,   "Laplace transforms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066340/n066340100.png" /> conjugate systems"  ''Dokl. Akad. Nauk SSSR'' , '''71''' :  3  (1950)  pp. 437–439  (In Russian)</TD></TR></table>