Isometric embedding via strongly symmetric positive systems

Gui Qiang Chen, Jeanne Clelland, Marshall Slemrod, Dehua Wang, Deane Yang

Research output: Contribution to journalArticle

Abstract

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 (PDE), 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.

Original languageEnglish (US)
Pages (from-to)1-40
Number of pages40
JournalAsian Journal of Mathematics
Volume22
Issue number1
DOIs
StatePublished - 2018

Keywords

  • Isometric embedding
  • Strongly symmetric positive systems

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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