Net (in differential geometry)

A system $ \Sigma _ {n} = \{ \sigma ^ {1} \dots \sigma ^ {n} \} $ of $ n $ families $ ( n \geq 2) $ of sufficiently smooth curves in a domain $ G $ of an $ n $- dimensional differentiable manifold $ M $ such that: 1) through each point $ x \in G $ there passes exactly one curve of each family $ \sigma ^ {i} $; and 2) the tangent vectors to these curves at $ x $ form a basis for the tangent space $ T _ {x} $ to $ M $ at $ x $. The tangent vectors to the curves of one family $ \sigma ^ {i} $ belong to a $ 1 $- dimensional distribution $ \Delta _ {1} ^ {i} $ defined in $ G $. 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 $ \sigma ^ {i} $ are the integral curves of $ \Delta _ {1} ^ {i} $. A net $ \Sigma _ {n} \subset G $ is defined by specifying $ n $ one-dimensional distributions $ \Delta _ {1} ^ {i} $ such that the tangent space $ T _ {x} $ at each $ x \in G $ is the direct sum of the subspaces $ \Delta _ {1} ^ {i} $, $ i = 1 \dots n $. A net $ \Sigma _ {n} \subset G $ defines in $ G $ $ ( n - 1) $- dimensional distributions $ \Delta _ {n - 1 } ^ {i} $ such that at each point $ x \in G $ the subspace $ \Delta _ {n - 1 } ^ {i} ( x) \subset T _ {x} $ is the direct sum of the $ n - 1 $ one-dimensional subspaces $ \Delta _ {1} ^ {j} ( x) $, $ j \neq i $. The following types of nets are distinguished: holonomic nets, partially holonomic nets, for which certain of the $ \Delta _ {n - 1 } ^ {i} $ 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 $ \Delta _ {n - 1 } ^ {i} $ are non-integrable.

If a distribution $ \Delta _ {n - 1 } ^ {i} $, $ n > 2 $, is integrable and $ \gamma ^ {i} $ is an integral curve of $ \Delta _ {1} ^ {i} $, then through each point $ x \in \gamma ^ {i} $ there passes an integral manifold of $ \Delta _ {n- 1 } ^ {i} $ that carries a net $ \Sigma _ {n - 1 } ( x) $ of curves belonging to the families $ \sigma ^ {j} $, $ j \neq i $.

A net $ \Sigma _ {n} \subset G $ can also be defined by one of the following means: a) by a system of vector fields $ X _ {i} \subset \Delta _ {1} ^ {i} $; b) by a system of differential $ 1 $- forms $ \omega ^ {i} $ such that $ \omega ^ {i} ( X _ {j} ) = \delta _ {j} ^ {i} $; or c) by the field of an affinor $ \Phi $ such that $ \Phi ^ {n} = E $( $ E $ 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 $ \Sigma _ {n} $ in a space $ M $ with an affine connection $ \nabla $ is called geodesic if all its curves are geodesic. If a Riemannian manifold $ M $ with a torsion-free connection in which the metric tensor is covariantly constant carries an orthogonal Chebyshev net of the first kind, then $ M $ 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 $ E $. For example, suppose that a net $ \Sigma _ {n} $ defined in a domain $ G $ on a smooth surface $ V _ {n} $ in $ ( n+ k) $- dimensional projective space $ ( k \geq 1) $ is conjugate, that is, at each point $ x \in G $ the directions $ \Delta _ {1} ^ {i} ( x) $, $ \Delta _ {1} ^ {j} ( x) $ of the tangents to any two curves of $ \Sigma _ {n} $ through $ x $ are conjugate (two directions are conjugate if each belongs to the characteristic of the tangent plane $ T _ {x} $ when it is displaced in the other direction). If $ V _ {n} $ is not contained in a projective space of dimension less than $ n+ k $, then for $ k = 1 $, $ V _ {n} $ carries an infinite set of conjugate nets; for $ k = 2 $ the surface carries, in general, a unique conjugate net, but there are $ n $- dimensional surfaces on which there are no conjugate nets; for $ k > 2 $ only $ n $- dimensional surfaces of a special structure carry a conjugate net. For $ n > 2 $ a conjugate net need not be holonomic (see [3]). A special case of a holonomic conjugate net is an $ n $- conjugate system: A net $ \Sigma _ {n} $ 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 $ n+ k $ for $ n \geq 2 $, $ k \geq 0 $. Surfaces $ V _ {n} $ that carry an $ n $- conjugate system in the $ ( n + k) $- dimensional projective space, when $ k \geq n $, and for which at each point $ x \in V _ {n} $ the osculating space (the space of second differentials of the point $ x $) has dimension $ 2n $, were first considered by E. Cartan [4] under the name "manifolds of special projective type" (Cartan surfaces). The concept of the Laplace transformation (in geometry) was extended to such nets (see [5], [6]).

In the study of diffeomorphisms of nets, in terms of known properties of a net $ \Sigma _ {n} \subset M $ one describes properties of the net $ \phi ( \Sigma _ {n} ) \subset N $ for a given diffeomorphism $ \phi : M \rightarrow N $( for example, under a bending deformation or under a conformal mapping of a surface carrying a net), or one looks for a diffeomorphism $ \phi $ that preserves certain properties of $ \Sigma _ {n} $. For example, a net $ \Sigma _ {2} $ 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 is rhombic.

References

[1]	A.P. Norden, "Spaces with an affine connection" , Nauka , Moscow-Leningrad (1976) (In Russian)
[2]	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
[3]	V.T. Bazylev, "Multidimensional nets and their transformations" Itogi Nauk. Geom. 1963 (1965) pp. 138–164 (In Russian)
[4]	E. Cartan, "Sur les variétés de courbure constante d'un espace euclidien ou non euclidien" Bull. Soc. Math. France , 47 (1919) pp. 125–160
[5]	S.S. Chern, "Laplace transforms of a class of higher dimensional varieties in a projective space of dimensions" Proc. Nat. Acad. Sci. USA , 30 (1944) pp. 95–97
[6]	R.V. Smirnov, "Laplace transforms of conjugate systems" Dokl. Akad. Nauk SSSR , 71 : 3 (1950) pp. 437–439 (In Russian)