Hyperbolic PDE (like the wave equation) are roughly speaking, PDE that satisfy the “finite propagation speed of information” property. They are ubiquitous in mathematical physics (essentially, most fundamental laws of nature are hyperbolic). 


However, do hyperbolic PDE occur in any other areas of mathematics that do not have ties to the real world ? 



Hyperbolic PDE in mathematics
Hyperbolic PDEs arise unexpectedly in some differential geometric questions involving prescribed data. What's weird in these cases is that there is no natural time coordinate in the PDEs. Here are some examples:


1. Bryant; Griffiths; Yang.
Characteristics and existence of isometric embeddings. 
Duke Math. J. 50 (1983), no. 4, 893–994.
2. DeTurck, Yang.
Existence of elastic deformations with prescribed principal strains and triply orthogonal systems. 
Duke Math. J. 51 (1984), no. 2, 243–260. As an aside, the triply orthogonal system result implies the local existence of coordinates on a Riemannian 3-manifold for which the metric tensor is diagonal. This generalizes isothermal coordinates on a Riemannian 2-manifolds.


