The terminology involved in this post comes from the book on smooth manifolds written by John M. Lee. By the Whitney embedding theorem, every smooth $n$-manifold $M$ admits a proper smooth embedding into $\mathbb{R}^{2n+1}$. And Professor Lee then concludes that $M$ is diffeomorphic to a properly embedded submanifold of $\mathbb{R}^{2n+1}$. I have no question about why $M$ is diffeomorphic to an embedded submanifold of $\mathbb{R}^{2n+1}$, which is just a direct application of Whitney's theorem. But somehow I can't understand the reason why the submanifold is properly embedded. In order for this to come true, one must be able to show that the inclusion map $\iota:F(M)\hookrightarrow\mathbb{R}^{2n+1}$ is a proper map, where $F$ is the diffeomorphism. Does anyone have an idea? Thank you.
Every smooth $n$-manifold is diffeomorphic to a properly embedded submanifold of $\mathbb{R}^{2n+1}$.
differential-geometrysmooth-manifolds
Best Answer
Consider a smooth embedding $F:M\hookrightarrow \Bbb R^{2n+1}$ with $F$ proper, and the inclusion $\iota:F(M)\hookrightarrow \Bbb R^{2n+1}$. Let $φ:M\xrightarrow{≃}F(M)$ be the diffeomorphism obtained from $F$ restricting the codomain. So, $F=\iota\circ φ$ i.e., $F\circ φ^{-1}=\iota$, i.e., $\iota$ is a composition of two proper maps, hence itself proper.