Geometry of Bäcklund Transformations II: Monge-Ampère Invariants

Yuhao Hu

Published 2019-04-04Version 1

Classically, given a surface $S\subset \mathbb{E}^3$ with constant Gauss curvature $K<0$, one can generate a 1-parameter family of surfaces with the Gauss curvature $K$ by solving only ODEs. This is an example of a rank-$1$ B\"acklund transformation relating two hyperbolic Euler-Lagrange systems. Our paper is motivated by the natural question: For which pairs of hyperbolic Euler-Lagrange systems does there exist a rank-$1$ B\"acklund transformation relating them? We express some obstructions to such existence in terms of the local invariants of the Euler-Lagrange systems. In addition, we discover a class of B\"acklund transformations relating two hyperbolic Euler-Lagrange systems of distinct types.