# Nash embedding with target which is not $\mathbb{R}^{m}$

I'm curious about the following question:

Given $(M^n,g)$ a closed Riemannian manifold, is there always a $C^\infty$ isometric embedding $F:(M^n,g) \to (\mathbb{S}^{m},g_{std})$ for $m$ large enough?

Remarks:

• If we replace $(\mathbb{S}^{m},g_{std})$ by $(\mathbb{R}^m,\delta)$ for $m$ large enough the answer is "yes" by the $C^\infty$-Nash Embedding Theorem.

• Alternatively, if you replace $C^\infty$ by $C^1$, I think the answer is also "yes" and one can simply use the same h-principle proof as for the Nash--Kupier theorem (this is claimed somewhere in Eliashberg and Mišačev's book on the h-principle, but I don't remember the exact place).

Closely related questions that I'd be curious to know about are (arranged in increasing order of difficulty, with the original question between 1 and 2):

1. Is this true if we relax the target to be a sphere of any radius?

2. Is there an integer $N = N(n)$ and constant $D > 0$ so that if $diam(M^n,g) \leq 1$ then for any $N(n)$-dimensional manifold of diameter at least $D$, $(V^N,g_V)$ we can always find an isometric embedding of $M$ into $V$?

3. What happens to the last question if we drop the diameter restrictions?

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