The isometric embedding theorems of John F. Nash

This thesis is based on two masterpieces of John Nash in pure mathematics, but in a re-elaborate version written by Camillo De Lellis in The masterpieces of John Forbes Nash Jr. In particular, the main focus of this work is the isometric embedding problem for Riemannian manifolds. I treat two cases: the first one is related to the C^1 isometric embeddings, while the second one regards the smooth isometric embeddings. Firstly, we consider a Riemannian manifold and a short map: we prove that there exists a C^1 isometric immersion in R^N, for some N, which is close enough to the short map. In order to achieve this, we use some basic facts on differential geometry and an iterative argument based on producing a sequence of short immersions that converges to the desired one. In the second case we consider a C^k Riemannian manifold and we want to prove the existence of a C^k isometric embedding. To do this we start by applying the first case in order to find a C^1 isometric embedding, then we smoothen it and deform it with a smooth embedding and finally apply a perturbation theorem which permits us to find the right isometric embedding.