5
$\begingroup$

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?

$\endgroup$
1
  • $\begingroup$ From my point of view: if we defined a manifold to be a subset of some euclidean space, then anytime we said "this object $X$ is a manifold" we would have to exhibit a description of $X$ as a subset of some $\mathbb{R}^N$. From what I understand about Whitney's Theorem, this might not be so easy to do explicitly... $\endgroup$ Aug 19, 2022 at 8:49

1 Answer 1

8
$\begingroup$

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.

$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .