In this talk, we will discuss an infinite dimensional Riemannian geometric approach to the nonlinear wave equations and how this perspective provides additional insight into the properties of their solutions. To this end, we will first review finite dimensional Riemannian geometry and gently introduce the infinite dimensional version. Then we will reveal some concrete geometric interpretations of the Hunter-Saxton partial differential equation. It is inevitable to use vocabulary from differential geometry and functional analysis. But I will make sure to explain intuitive ideas with examples and maintain our focus on understanding the mathematical perspective rather than technicalities.