Local smooth isometric embeddings of low-dimensional Riemannian manifolds into Euclidean spaces

Abstract

Local smooth isometric embedding problems of low dimensional Riemannian manifolds into Euclidean spaces are studied. Namely, we prove the existence of a local smooth isometric embedding of a smooth Riemannian 3 3 -manifold with nonvanishing curvature into Euclidean 6 6 -space. For proving this, we give a local solvability theorem for a system of a nonlinear PDE of real principal type. To obtain the local solvability theorem, we need a tame estimate for the linearized equation corresponding to the given PDE, which is presented by two methods. The first is based on the result of Duistermaat-Hörmander which constructed the exact right inverse for linear PDEs of real principal type by using Fourier integral operators. The second method uses more various properties of Fourier integral operators given by Kumano-go, which seems to be a simpler proof than the above.