# Conformal geometry of surfaces in the Lagrangian--Grassmannian and second order PDE

```@inproceedings{The2010ConformalGO,
title={Conformal geometry of surfaces in the Lagrangian--Grassmannian and second order PDE},
author={D. The},
year={2010}
}```
• D. The
• Published 2010
• Mathematics
Of all real Lagrangian–Grassmannians LG(n, 2n), only LG(2, 4) admits a distinguished (Lorentzian) conformal structure and hence is identified with the indefinite Möbius space S. Using Cartan’s method of moving frames, we study hyperbolic (timelike) surfaces in LG(2, 4) modulo the conformal symplectic group CSp(4,R). This CSp(4,R)-invariant classification is also a contact-invariant classification of (in general, highly non-linear) second order scalar hyperbolic PDE in the plane. Via LG(2, 4… Expand

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#### References

SHOWING 1-10 OF 23 REFERENCES
Integrable GL(2) Geometry and Hydrodynamic Partial Differential Equations
This article is a local analysis of integrable GL(2)-structures of degree 4. A GL(2)-structure of degree n corresponds to a distribution of rational normal cones over a manifold M of dimension (n+1).Expand
Contact Geometry of Hyperbolic Equations of Generic Type
• D. The
• Mathematics, Physics
• 2008
We study the contact geometry of scalar second order hyperbolic equations in the plane of generic type. Following a derivation of parametrized contact-invariants to distinguish Monge–Ampère (classExpand
Contact geometry of multidimensional Monge-Ampere equations: characteristics, intermediate integrals and solutions
• Mathematics
• 2010
We study the geometry of multidimensional scalar 2 order PDEs (i.e. PDEs with n independent variables) with one unknown function, viewed as hypersurfaces E in the Lagrangian Grassmann bundle M (1)Expand
Complete systems of invariants for rank 1 curves in Lagrange Grassmannians
Curves in Lagrange Grassmannians naturally appear when one studies intrinsically ”the Jacobi equations for extremals”, associated with control systems and geometric structures. In this way oneExpand
Parametrized curves in Lagrange Grassmannians
• Mathematics
• 2007
Abstract Curves in Lagrange Grassmannians naturally appear when one studies Jacobi equations for extremals, associated with geometric structures on manifolds. We fix integers d i and consider curvesExpand
On the integrability of symplectic Monge–Ampère equations
• Mathematics, Physics
• 2010
Abstract Let u be a function of n independent variables x 1 , … , x n , and let U = ( u i j ) be the Hessian matrix of u . The symplectic Monge–Ampere equation is defined as a linear relation amongExpand
Geometric Aspects of Second-Order Scalar Hyperbolic Partial Differential Equations in the Plane
Geometric Aspects of Second-Order Scalar Hyperbolic Partial Differential Equations in the Plane by Martin Juras, Doctor of Philosophy Utah State University, 1997 Major Professor: Dr. Ian M. AndersonExpand
Integrable Equations of the Dispersionless Hirota type and Hypersurfaces in the Lagrangian Grassmannian
• Mathematics
• 2007
We investigate integrable second-order equations of the formwhich typically arise as the Hirota-type relations for various (2 + 1)-dimensional dispersionless hierarchies. Familiar examples includeExpand
Characteristics and the Geometry of Hyperbolic Equations in the Plane
• Mathematics
• 1993
Abstract This paper focuses on the implementation and application of Elie Cartan′s ideas on characteristics for second order non-linear hyperbolic equations in the plane. The topics include aExpand
Conformal Differential Geometry and Its Generalizations
• Mathematics
• 1996
Conformal and Pseudoconformal Spaces. Hypersurfaces in Conformal Spaces. Submanifolds in Conformal and Pseudoconformal Spaces. Conformal Structures on a Differentiable Manifold. The Four--DimensionalExpand