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I learnt from the Nash Embedding theorem that every Riemannian manifold can be isometrically embedded into some Euclidean space. But is there some pratical way to construct the embedding? A toy example, suppose the metric of the manifold is g=m*diag(3), where m varies over the manifold. How to embed it into Euclidean space

Baoquan Feng
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  • Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking. – Community Mar 29 '24 at 08:11
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    Nash's theorem is a hard existence theorem for a solution of some nonlinear PDE. Typically, there is no way to "practically construct" a solution whatever you mean by this. One can find approximate solutions but that's all. Two more things: (1) You should use MathJax to write main in your questions. (2) Metrics that you tried to describe are called conformally flat. – Moishe Kohan Mar 29 '24 at 12:43
  • Maybe not a duplicate, but related: https://math.stackexchange.com/questions/4780961/alternate-definition-of-riemannian-manifold – Andrew D. Hwang Mar 29 '24 at 13:54

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