On a remark of John Lee concerning smooth manifolds

Question

In John Lee's book Introduction to Smooth Manifolds, on the first page of the first chapter, he writes:

..."But for more sophisticated applications it is an undue restriction to require smooth manifolds to be subsets of some ambient Euclidean space."

But on page 134 of the same book he proves the Whitney Embedding theorem:

Every smooth $n$-manifold with or without boundary admits a proper smooth embedding into $\mathbb{R}^{2n+1}$.

which seems to me to directly contradict his statement on the first page.

Question: Am I missing something?

Answer 1

It is true that each manifold can be embedded into some Euclidean space $\mathbb R^N$. In other words, each manifold is dffeomorphic to a a submanifold of a Euclidean space, and therefore one could argue that it is sufficient to develop a theory for such submanifolds. In fact, some concepts (for example, the tangent space) even allow a more intuitive access for submanifolds than for "abstract" manifolds.

However, the embedding theorem is an existence theorem which does not provide a canonical embedding. There are many such embeddings, and each depends on certain choices. A priori it is not even clear what the minimal dimension $N$ of an ambient $\mathbb R^N$ is.

Many well-known manifolds are not given as submanifold, but by other constructions. Here are some examples:

1. Projective spaces $\mathbb RP^n$ and $\mathbb CP^n$

2. More generally quotients of manifolds by group actions.

3. Grassmann manifolds and Stiefel manifolds.

4. Quotients like $[0,1]/(0 \sim 1)$.

Try to find explicit embeddings of these objects into a Euclidean space - you will see it is not easy.

Thus I completely agree to the statement that

it is an undue restriction to require smooth manifolds to be subsets of some ambient Euclidean space.