Let \(M\) be a manifold. A Goursat structure over \(M\) is a natural generalization of contact structures and Engel structures. It is given by a 2-rank vector distribution \({\mathcal D}\) (smooth 2-dimensional subbundle of the tangent bundle \(TM\)) which for each point \(p \in M\) satisfies \(\dim {\mathcal D}^{(k)} M(p) = k + 2\), where the sequence \({\mathcal D} = {\mathcal D}^{(0) }\subset {\mathcal D}^{(1)} \subset \ldots \subset{\mathcal D}^{(n-2)} = TM\) is inductively defined by \[ {\mathcal D}^{(k+1)} = {\mathcal D}^{(k)} + [ {\mathcal D}^{(k)},{\mathcal D}^{(k)}]. \] The growth vector at a point \(p\) is given by the sequence \(\left(\dim {\mathcal D}_k(p)\right)_{k\in\{ 0,\ldots,r\}}\), where the modules \({\mathcal D}_k\) are also defined inductively by \[ {\mathcal D}_0 = {\mathcal D} \text{ and } {\mathcal D}_{k +1} = {\mathcal D}_k + [{\mathcal D}_0,{\mathcal D}_k]. \] E. v. Weber had previously introduced the dual 2-rank Pfaffian system \({\mathcal I}\) (i.e., a smooth two-dimensional subbundle of the cotangent bundle \(T^\star M\)) which in fact corresponds to \({\mathcal D}^\perp\) [“Zur Invariantentheorie der Systeme Pfaff’scher Gleichungen”, Leipz. Ber. 50, 207-229 (1898; JFM 29.0302.01)]. This class of systems was extensively studied at a generic point \(p\) by É. Cartan, who called them “systems of class zero” [“Sur l’équivalence absolue de certins systèmes d’equations differentielles et sur certaines familles de courbes”, Bull. Soc. Math. Fr. 42, 12-48 (1914; JFM 45.1294.04)] and by E. Goursat [“Leçons sur le problème de Pfaff”, Hermann, Paris (1923)].
In the main theorem of the article under review, it is proved (in the authors’ words) that ‘the growth vector is not the only local invariant for a Goursat structure on \(\mathbb{R}^n, n \geq 9\), and also that there exist infinitely many locally non-equivalent Goursat structures on \(\mathbb{R}^n, n \geq 10\).’
Previous fundamental work was done by A. Kumpera and C. Ruiz [An. Acad. Bras. Cienc. 55, 225-229 (1983; Zbl 0529.58002)] and D. Levebvre and M.-T. Pourprix [J. Lond. Math. Soc., (2) Ser. 29, 367-379 (1984; Zbl 0529.58004)]. One of the authors, P. Mormul, has also worked along this line with Fernand Pelletier [P. Mormul and F. Pelletier, Bull. Pol. Acad. Sci., Math. 45, No. 4, 399-418 (1997; Zbl 0899.58003) and C. R. Acad. Sci., Paris, Sér. I 322, No. 9, 865-868 (1996; Zbl 0865.58001)].
This paper includes a good review of the main results on Goursat structures along with references.