Whitney approximation for manifold-valued maps

Choose a proper embedding $j:N\hookrightarrow {\mathbb{R}}^m$ from [L1]. By [L3], the embedded image $j(N)$ has a tubular neighbourhood $U$ with smooth retraction $r:U\to j(N)$.

Apply the fine-approximation lemma from [L2] to the continuous map $j\circ F:M\to {\mathbb{R}}^m$ and the tubular neighbourhood $U$, obtaining a positive continuous error function $\epsilon$ whose $\epsilon (p)$-ball around $j(F(p))$ lies in $U$. Then use the Euclidean Whitney theorem from [L2] to obtain a smooth map $H:M\to {\mathbb{R}}^m$ with $\mathrm{\parallel }H(p)-j(F(p))\mathrm{\parallel }<\epsilon (p)$ for all $p$, hence with image in $U$.

Define $\tilde{F}:=j^{-1}\circ r\circ H\mathrm{.}$ This map is smooth. For each $p\in M$ and $t\in I$, the point $(1-t)j(F(p))+tH(p)$ stays in the same $\epsilon (p)$-ball around $j(F(p))$, hence stays in $U$ by step 2.1. Since $r$ fixes $j(N)$ pointwise, the formula $(p,t)\mapsto j^{-1}(r((1-t)j(F(p))+tH(p)))$ is therefore well defined and continuous on $M\times I$, and it gives a homotopy from $F$ to $\tilde{F}$ in the sense of [L3].