Why focus only on geometry? Nash investigated a lot of fields, in only thirteen (?) published papers. That on the isometric embedding contains a fixed point theorem that is now called the Nash-Moser iteration, which turns out to be extremely fruitful in many parts of analysis, especially in PDEs, when ordinary methods fail. More precisely, it is used when the linearization about a given solution yields small divisors troubles.
Nash also had a seminal work on compressible viscous fluid flows.
Edit. As mentionned by Pedro, one speaks of the Nash-Moser implicit function theorem. Nevertheless, the method is really a (far reaching) adaptation of Newton's iteration (as Pedro quote it). Isn't Newton method a fixed point algorithm designed to find and approach zeroes of functions ?