Methods and Applications of Analysis

Volume 26 (2019)

Number 1

In Memory of Professor John N. Mather: Part 3 of 3

Guest Editors: Sen Hu, University of Science and Technology, China; Stanisław Janeczko, Polish Academy of Sciences, Poland; Stephen S.-T. Yau, Tsinghua University, China; and Huaiqing Zuo, Tsinghua University, China.

$G_2$-geometry in contact geometry of second order

Pages: 65 – 100



Keizo Yamaguchi (Department of Mathematics, Faculty of Science, Hokkaido University, Sapporo, Japan)


In [13], [14], [15], we formulate the contact equivalence of systems of second order partial differential equations for a scalar function as the Contact Geometry of Second Order or as the geometry of $PD$-manifolds of second order, generalizing works [3], [4] of E. Cartan. Especially, in [13], generalizing the famous $G_2$-models in [3], we observed, for each exceptional simple Lie algebra $X_{\ell}$, we could find the overdetermined system ($A_{\ell}$) and the single equation of Goursat type $B_{\ell}$, whose symmetry algebras are isomporphic with $X_{\ell}$ and formulated this fact as the $G_2$-geometry. The main purpose of the present paper is to construct the (local) models for overdetermined systems $A_{\ell}$ explicitly for each exceptional simple Lie algebra and also for the classical type analogy for $BD$ type. We will also give parametric descriptions of the single equation of Goursat type ($B_{\ell}$). Our constructions are based on the explicit calculation, in terms of Chevalley basis, of the structure of the Goursat gradation of each exceptional simple Lie algebra and each simple Lie algebra of $BD$ type.


$G_2$-geometry, contact transformations, Goursat gradation of exceptional simple Lie algebras

2010 Mathematics Subject Classification

53C15, 58A15, 58A20, 58A30

This work was partially supported by the grant 346300 for IMPAN from the Simon Foundation and the matching 2015-2019 Polish MNiSW fund.

Received 14 May 2018

Accepted 22 August 2019

Published 14 November 2019