Applying Nash-Moser in Differential Topology

Abstract

The h-principle is a fruitful and active area of research with applications in many areas of differen- tial geometry. Its main goal is to prove that the space of various geometric structures is (weakly) homotopy equivalent to more manageable spaces. There are various ways to obtain such results; from these we will introduce Eliashberg and Mishachev’s famous holonomic (R)-approximation theorem. Nash-Moser theorems are a powerful class of theorems generalizing the inverse function theorem to spaces more difficult than Banach spaces, with wide applications in PDEs and differential geometry. They originate from Nash’s seminal paper on isometric Riemannian embeddings. The method of proof was later adapted by Moser into a more general theorem which he subsequently used to solve various PDEs. Many other versions have since arisen; in this thesis we treat versions by Gromov and Hamilton. Motivated by the problem of finding an h-principle for isotropic immersions into almost symplectic manifolds, we show how the Nash-Moser theorem may be used hand in hand with holonomic approximation to obtain h-principles for a certain kind of differential relation.