Throughout this post we shall always work in the smooth category, thus all manifolds, maps, coordinate charts, and functions are assumed to be smooth unless explicitly stated otherwise.

A (real) manifold $latex {M}&fg=000000$ can be defined in at least two ways. On one hand, one can define the manifold *extrinsically*, as a subset of some standard space such as a Euclidean space $latex {{bf R}^d}&fg=000000$. On the other hand, one can define the manifold *intrinsically*, as a topological space equipped with an atlas of coordinate charts. The fundamental *embedding theorems* show that, under reasonable assumptions, the intrinsic and extrinsic approaches give the same classes of manifolds (up to isomorphism in various categories). For instance, we have the following (special case of) the Whitney embedding theorem:

**Theorem 1 (Whitney embedding theorem)** Let $latex {M}&fg=000000$ be a compact manifold. Then there exists an embedding $latex {u: M rightarrow {bf…