## Hokkaido Mathematical Journal

- Hokkaido Math. J.
- Volume 41, Number 3 (2012), 409-440.

### Geometric characterization of Monge-Ampère equations

#### Abstract

It is well known that a Monge-Ampère equation can be expressed in terms of exterior differential system—Monge-Ampère system, which is the ideal generated algebraically by a contact form and a 2-form and its exterior derivatives on a 5-dimensional contact manifold, and the system is independent of the choice of coordinate system. On the other hand, a single second order partial differential equation of one unknown function with two independent variables corresponds to the differential system on a hypersurface of Lagrange-Grassmann bundle over a 5-dimensional contact manifold obtained by restricting its canonical system to the hypersurface. We observe relations between Monge characteristic systems of Monge-Ampère equation and those of Monge-Ampère system and particularly analyze structure equations of those systems. This observation leads to the result—to characterize Monge-Ampère equation by the property that the certain differential system defined from the Monge characteristic system drops down to the contact manifold.

#### Article information

**Source**

Hokkaido Math. J., Volume 41, Number 3 (2012), 409-440.

**Dates**

First available in Project Euclid: 24 October 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.hokmj/1351086222

**Digital Object Identifier**

doi:10.14492/hokmj/1351086222

**Mathematical Reviews number (MathSciNet)**

MR3012457

**Zentralblatt MATH identifier**

1256.58001

**Subjects**

Primary: 58A15: Exterior differential systems (Cartan theory)

Secondary: 53D10: Contact manifolds, general

**Keywords**

differential system exterior differential system partial differential equation Monge-Ampere equation Goursat equation Monge characteristic system

#### Citation

YANO, Atsushi. Geometric characterization of Monge-Ampère equations. Hokkaido Math. J. 41 (2012), no. 3, 409--440. doi:10.14492/hokmj/1351086222. https://projecteuclid.org/euclid.hokmj/1351086222