Given a riemmanian manifold (M,g) (of a given smoothness) is it possible to isometrically embed it (again with some smoothness) into a Euclidean space of sufficiently high dimension? This problem was solved affirmatively by Nash for the case that the metric tensor is continuous and the embedding is C^1 in 1954 and then for the case of C^r isometric embeddings of C^r Riemannian manifolds (r>3)in 1956. I will give some background to this problem and then focus on the solution to the C^1 case, attempting to highlight the stark difference between C^1 isometric embeddings and smoother ones. Finally, I will try to give some overview of the general analytical machinery, which replaced his more elementary geometric methods in the first paper, that Nash developed for the general case.
Graduate Student Geometry-Topology Seminar
Wednesday, November 15, 2006 - 11:00am
Daniel Pomerleano
University of Pennsylvania