Les systèmes de Pfaff, à cinq variables et les équations aux dérivées partielles du second ordre

Abstract

Les systèmes de Pfaff, à cinq variables et les équations aux dérivées partielles du second ordre Annales scientifiques de l'É.N.S. 3 e série, tome 27 (1910), p. 109-192 <http://www.numdam.org/item?id=ASENS_1910_3_27__109_0> © Gauthier-Villars (Éditions scientifiques et médicales Elsevier), 1910, tous droits réservés. L'accès aux archives de la revue « Annales scientifiques de l'É.N.S. » (http://www. elsevier.com/locate/ansens) implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ ' h).

Cartan’s incomplete classification and an explicit ambient metric of holonomy $$\mathrm{G}_2^*$$

“…They are locally equivalent (near u ∈ M and u ∈ M ) if there are neighborhoods U of u and U of u such that (U, D| U ) and (U , D | U ) are equivalent via a diffeomorphism that maps u to u . The geometry of these structures was first studied systematically by Cartan, in his well-known "Five Variables" article [14]. There, he solved the equivalence problem for this geometry by canonically assigning to each such distribution (M, D) a principal Q-bundle E → M and a g * 2 -valued pseudoconnection ω on E, where Q is a particular parabolic subgroup of G * 2 .…”

“…In [14] Cartan claimed to classify, up to local equivalence, and implicitly in the complex setting, all (2, 3, 5) distributions whose infinitesimal symmetry algebra has dimension at least 6 and which have constant root type: The fundamental curvature invariant of a (2, 3, 5) distribution (M, D)-analogous to the Riemann curvature tensor in Riemannian geometry-is a section A ∈ ( 4 D * ). The value of A at u ∈ M depends on the 4-jet of D at u, and the quantity A is natural in the sense that it is preserved by diffeomorphism.…”

“…Cartan's most involved application of his powerful method of equivalence was the resolution of the (local) equivalence problem for the geometry of 2-plane distributions D on 5-manifolds M, which he reported in his landmark 1910 "Five Variables" paper [14]. Such distributions that are maximally nonintegrable in the sense that This geometry has proved interesting for numerous reasons: It comprises a first class of distributions with continuous local invariants, it is connected to the geometry of surfaces rolling on one another without slipping or twisting [5,8,10] and hence to natural problems in control theory [2], it is intimately linked (as Cartan showed in the aforementioned article 1 ) to the exceptional simple complex Lie group of type G C 2 and then via the natural inclusion G * 2 → SO(3, 4) to conformal geometry [26], and it is the appropriate structure for naturally projectively compactifying strictly nearly (para-)Kähler structures in dimension 6 [21].…”

On the equivalence problem and integration of differential systems

“…From this expression, we can get another integral generator (θ\ , θ s ) of S, where θ ι is given by This can be considered as a standard form of generators of S. Using such a standard form, we can often construct integral manifolds of S (see [3] p. 159 and pp. 169-171).…”

“…The following lemma is well-know τ n (see [4] p. 1060 and [5] p. 52). In this case, Ch (S') is also a covariant system of S. In [3] and [4], https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0027763000000878 E. Cartan used such covariant systems for the problem of integration of second order partial differential equations.…”