I am trying to understand Matthias Günther's proof of Nash's embedding theorem which is outlined in

Günther, Matthias. Isometric embeddings of Riemannian manifolds. Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990), 1137–1143, Math. Soc. Japan, Tokyo, 1991.

There is a more detailed version available in German:

Günther, Matthias. Zum Einbettungssatz von J. Nash. (German) [On the embedding theorem of J. Nash] Math. Nachr. 144 (1989), 165–187.

The following quote is form the Kyoto-proceedings:

Theorem 3. Let $g$ be of class $C^\infty$, let $u_0\in C^\infty(M, \mathbb R^q)$ be a free embedding with $q\leq n(n + 3)/2 + 5$ and $d_{u_0}\cdot d_{u_0} < g$. Further, let $\delta$ be any positive continuous function on $M$. Then there exists an embedding $u\in C^\infty(M, \mathbb R^q)$ such that $du\cdot du = g$ and $|u(x) — u_0(x)| < \delta(x)$ for every $x\in M$.

Concerning the existence of free embeddings $u_0$ we have the following proposition. Its proof is based on the well known theorem of Sard, see e.g. Gromov and Rohlin (1970, Sect. 2.5). The condition $d_{u_0}\cdot d_{u_0} < g$ can be reached by easy manipulations.

Proposition. If $g$ is a continuous metric on $M$, then there exists a free embedding $u_0 \in C^\infty(M, \mathbb R^q)$ with $q = n(n + 5)/2$ and $d_{u_0}\cdot d_{u_0} < g$ on $M$.

My question is: Can someone tell me, what these "easy manipulations" are, that ensure that the induced metric is less than $g$? There is also the book "Isometric Embedding of Riemannian Manifolds in Euclidean Spaces" by Han and Hong, where they proof the proposition in detail but only for the case where $M$ is compact. But then it is obviously no problem to rescale the embedding appropriately without destroying the freeness. But how is this done in the non-compact case?


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