Author
Chen, G
Clelland, J
Slemrod, M
Wang, D
Yang, D
Journal title
Asian Journal of Mathematics
DOI
10.4310/AJM.2018.v22.n1.a1
Issue
1
Volume
22
Last updated
2024-03-26T22:48:00.25+00:00
Page
1-40
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.
Symplectic ID
657179
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Publication type
Journal Article
Publication date
10 May 2018
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