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 - Jan 1 2018

Fingerprint

Positive Systems
Isometric Embedding
Local Existence
Partial differential equations
Ordinary differential equations
Microlocal Analysis
Tensors
Curvature Tensor
Systems of Partial Differential Equations
Singular Point
System of Ordinary Differential Equations
Positivity
Existence Theorem
Existence Results
Riemannian Manifold
Euclidean space
Partial differential equation
Theorem

Keywords

  • Isometric embedding
  • Strongly symmetric positive systems

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

Cite this

Isometric embedding via strongly symmetric positive systems. / Chen, Gui Qiang; Clelland, Jeanne; Slemrod, Marshall; Wang, Dehua; Yang, Deane.

In: Asian Journal of Mathematics, Vol. 22, No. 1, 01.01.2018, p. 1-40.

Research output: Contribution to journalArticle

Chen, Gui Qiang ; Clelland, Jeanne ; Slemrod, Marshall ; Wang, Dehua ; Yang, Deane. / Isometric embedding via strongly symmetric positive systems. In: Asian Journal of Mathematics. 2018 ; Vol. 22, No. 1. pp. 1-40.
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