0

I am trying to understand this answer on an isometric embedding of a sphere (with its geodesic distance) to an euclidean space cannot exist and Nash embedding theorem.

The statement of Nash's theorem is somewhat straightforward to follow which says - "every Riemannian manifold can be isometrically embedded as a submanifold of $\Bbb R^n$". Also, I am able to understand that the metric on $S^2$ is the induced metric from $\Bbb R^{3}$.

However, I am unable to follow these statements - "In that case, the identity map is a locally metric-preserving embedding into $\mathbb R^2$, but it doesn't preserve the global distance. .... Thus, the natural embedding works as an isometry when we view the two spaces as Riemannian manifolds, but not when we consider them directly as metric spaces. "

Can you please clarify the following: What is the identity map being referred here? How is it only locally isometric but not globally? What is the natural embedding?

P.S. - I am a beginner in this field and surely, not familiar with much of the advanced concepts. Requesting to help me in understand in an intuitive manner if possible.

honeybadger
  • 1,125
  • Borsuk-Ulam tells us that, given any continuous map $f \colon S^2\to\mathbb{R}^2$, there is a point $x\in S^2$ such that $f(x)=f(-x)$. Whatever the distance between $x$ and $-x$ is, that goes to $0$ after applying $f$. –  Mar 21 '20 at 05:35
  • @NelliKuukeri: But is that the identity map being referred here? – honeybadger Mar 21 '20 at 05:47
  • Where is this "here"? –  Mar 21 '20 at 06:32
  • @NelliKuukeri: In the question - "In that case, the identity map is a locally metric-preserving embedding into R2, but it doesn't preserve the global distance". I am specifically confused about the details of the identity map - meaning its domain, codomain, and exact functional form. – honeybadger Mar 21 '20 at 06:43
  • 2
    The linked answer, while highly upvoted, is non-rigorous. I suggest, you just ignore it. The correct statement requires you to understand what a Riemannian metric is and the notion of a Riemannian isometry. This would be discussed in any serious Riemannian Geometry textbook. – Moishe Kohan Mar 22 '20 at 05:22

0 Answers0