Contact manifolds, Lagrangian Grassmannians and PDEs

Abstract

In this paper we review a geometric approach to PDEs. We mainly focus on scalar PDEs in n independent variables and one dependent variable of order one and two, by insisting on the underlying (2n + 1)-dimensional contact manifold and the so-called Lagrangian Grassmannian bundle over the latter. This work is based on a Ph.D course given by two of the authors (G. M. and G. M.). As such, it was mainly designed as a quick introduction to the subject for graduate students. But also the more demanding reader will be gratified, thanks to the frequent references to current research topics and glimpses of higher-level mathematics, found mostly in the last sections.

Keywords

• contact and symplectic manifolds
• jet spaces
• lagrangian grassmannians
• first and second order pdes
• symmetries of pdes
• characteristics
• monge-ampère equations
• pdes on complex manifolds
• 32c15
• 35a30
• 35k96
• 53c30
• 53c55
• 53d05
• 53d10
• 58a20
• 58a30
• 58j70