Isometric embedding via strongly symmetric positive systems

Autor:	Chen, Gui-Qiang, Clelland, Jeanne, Slemrod, Marshall, Wang, Dehua, Yang, Deane
Rok vydání:	2015
Předmět:	Mathematics - Differential Geometry Mathematics - Analysis of PDEs
Druh dokumentu:	Working Paper
Popis:	We give a new proof for the local existence of a smooth isometric embedding of a smooth $3$-dimensional Riemannian manifold with nonzero Riemannian curvature tensor into $6$-dimensional Euclidean space. Our proof avoids the sophisticated arguments via microlocal analysis used in earlier proofs. In Part 1, we introduce a new type of system of partial differential equations, which is not one of the standard types (elliptic, hyperbolic, parabolic) but satisfies a property called strong symmetric positivity. Such a PDE system is a generalization of and has properties similar to a system of ordinary differential equations with a regular singular point. A local existence theorem is then established by using a novel local-to-global-to-local approach. In Part 2, we apply this theorem to prove the local existence result for isometric embeddings. Comment: 39 pages
Databáze:	arXiv
Externí odkaz:	http://arxiv.org/abs/1502.04356 Zobrazit plný text záznamu View this record from Arxiv