Differential and (Semi/Pseudo-)Riemannian geometry provide a framework for doing calculus on finite dimensional manifolds and have applications in physics (general relativity) and dynamical systems analysis (e.g. control theory). I've wondered how much work has been done to develop a calculus on infinite dimensional manifolds to aid in, say, bifurcation analysis for PDEs like the bifurcation problems in continuum mechanics as discussed in chapter 7 of Marsden and Hughes' Mathematical Foundations of Elasticity. Patrolling Google Scholar or Web of Science has not been enlightening in whetting my curiosity. I usually find references that leave me unsure if they're on topic or, if I can glean that they do seem to touch on this topic, they are beyond my current skill set to gather any understanding or sense of the field.
Apologies if it seems like my intuitions are off on what is and isn't possible or what has and hasn't been done in these fields. I'm more of an applied mathematician in training, but I'd like to use more geometry in my work on solid mechanics and other PDE analysis/simulation problems because, well, it's beautiful.