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Let $M$ be a compact smooth $n-$manifold. Suppose that $N \ge 2n$. Show that every smooth map from $M$ to $\mathbb{R}^N$ can be uniformly approximated by smooth immersions. This is a problem from John M Lee's book on Smooth Manifolds. I am using the following Corollary: (named 6.17 in the book)

Suppose that $M$ is a compact smooth $n$-manifold with or without boundary. If $N \ge 2n+1$, then every smooth map from $M$ to $\mathbb{R}^N$ can be uniformly approximated by emdeddings.

If $N \ge 2n+1$, then the claim follows by the above corollary. Let's assume that $N=2n$. Let $F:M \to \mathbb{R}^N$ be a smooth map. Consider $$i\circ F: M \to \mathbb{R}^{2n+1}$$ where $i:\mathbb{R}^N \to \mathbb{R}^{N+1}$ is the inclusion map. Again by the above Corollary, $i\circ F$ can be approximated by smooth immersions(injective). Now I want to project these smooth immersions to $\mathbb{R}^N$ along a line so that they still remain an immersion. I feel like using Sard's theorem, which tells me that $i\circ F(M)$ has measure zero(since dim$(M)=n \lt 2n+1$). Shall I just take any vector in ($\mathbb{R}^{2n+1} -(F(M)))$ and project along it??

Thanks for the help!!

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Assume $M$ is embedded in $\Bbb R^{2n+1}$. Consider the map $f_l:M \to H_l \cong \Bbb R^N$ which projects $M$ along $l$ onto the hyperplane at the origin $H_l$ given by taking the orthogonal complement of $l$. It is pretty easy to show that this map is an immersion at a point $p$ if and only if $l$ is not in $T_pM$.

Now from the embedding you get a map $F:S(TM) \to \Bbb RP^{2n}$ (here $S(TM)$ is the unit tangent bundle under some Riemannian metric) just by the identification of $TM$ to a subspace of $\Bbb R^{2n+1}$ coming from the embedding. $F$ will be surjective if and only if every line $l$ is in $T_pM$ for some $p$, so the problem reduces to showing $F$ is not surjective. Now the dimension of $S(TM)$ is $2n-1$, and the dimension of $\Bbb RP^{2n}=2n$, hence Sard's theorem finishes the proof (and some $f_l$ is an immersion everywhere).

To get this immersion to approximate a fixed map $f:M \to \Bbb R^{2n}$ use an embedding to approximate $(f,0): M \to \Bbb R^{2n+1}$ and pick your line arbitrarily close to $\{(0,0,\dots,t)| t \in \Bbb R\}$ using the fact that the complement of the image of $F$ is dense.

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