Weak Whitney embedding theorem

Question

I am currently reading Shigeyuki Morita's "Geometry of Differential Forms" and I have found one of his proofs to be a little bit on the difficult side. On page 36 he states the theorem " An arbitrary compact $\textit{C} \ ^\infty$ manifold M can be embedded into $R^N$ for sufficiently large N". The way he goes about proving the theorem goes as follows. He uses a finite collection of coordinate neighborhoods $\big\{(U_i,\phi_i)\big\}_{i=1}^r$ such that each $\phi_i:U\to R^n$ is an open disk $D(2)$ of radius 2 with center at the origin, and if we put $V_i=\phi_{i}^{-1}(D(1))$, $\big\{V_i\big\}_{i=1}^r$ is already a covering of M. Do not worry about the existence of such an atlas, it follows form the paracompactness of all manifolds and the fact that M itself is compact. Now he next defines for each $i$ a $C^\infty$ map $f_i:M\to S^n$ with the following two conditions: (i) the restriction of $f_i$ to $V_i$ is a diffeomorphism from $V_i$ onto the southern hemisphere $\big\{x\in S^n: x_{n+1} < 0\big\}$ of $S^n$ and (ii) $f_i$ maps the complement of $V_i$ to the northern hemisphere. He asks the reader to provide the map but I can't figure out how to construct such a map. A diffeomorphism from $V_i$ to the southern hemisphere is simple but I can't find a function which satisfies property (ii) as well as property (i). Any help is greatly appreciated!

Answer 1

Let $g$ be a smooth surjective mapping $D(2) \to \mathbb S^n$ so that (1) when restricted to $D(1)$, it is a diffeomorphism onto the southern hemisphere and $g(x)$ is the north pole when $\|x\| \ge c$ for some $c <1$. Then the mapping $ g\circ \phi_i : U_i \to \mathbb S^n$ can be extended to a map $f _i : M \to \mathbb S^n$ by setting $f(y) = \text{north pole}$ for all $y\in M\setminus U_i$. This map is smooth since $g\circ \phi$ maps all points in $\{x\in U_i: \|\phi_i(x) \|\ge c\}$ to the north pole.

To construct $g$, one way is to let $h :[-1, 1]\to [-1, 1]$ be a smooth non-decreasing map so that $h(t) = t$ for $t\in [-1, 0]$ and $h(t) =1$ for all $t\ge 3/4$. Let $H: \mathbb S^n \to \mathbb S^n$ be (write $\vec x = (x_1, \cdots, x_n)$) $$ H( \vec x, x_{n+1}) = \left( \sqrt{\frac{1-h(x_{n+1})^2}{1- x_{n+1}^2}} \vec x , h(x_{n+1})\right)$$

and let $g(x) = H(S(x)$, where $S$ is the stereographic projection (ntoe $S$ sends $D(1)$ to the southern hemisphere).