Whitney's Embedding Theorem

Question

The Whitney embedding theorem says that any smooth manifold of dimension $n$ may be embedded in $\mathbb{R}^{2n}$.

I am just beginning to study differential geometry for application to physics (general relativity), so I wonder why this result isn't quoted (though not, of course, proved) right at the outset in differential geometry texts.

Most texts launch right into abstract manifolds, but it seems to me that the Whitney theorem allows anyone interested in applications to restrict their study to Euclidean manifolds. This allows one to forego the abstract definition of tangent vectors as differential operators, but rather to use the conventional idea of vectors in $\mathbb{R}^n$ (or, rather, $\mathbb{R}^{2n}$!)for elements of the tangent space.

Also, one can use coordinate maps defined on the entire manifold, rather than restricting oneself to patches.

Am I off-base here? Would someone please share words of wisdom? Do I really have to study abstract manifold theory?

Answer 1

It is very tempting to only study submanifolds of numerical spaces instead of abstract manifolds but one should resist that temptation!
Yes, every manifold can be embedded in $\mathbb R^N$ but in a non canonical way and you won't (in general) be able to do any computations after the embedding.

This is a very common situation in mathematics:

$\bullet$ Every finite group is isomorphic to a subgroup of some permutation group $S_n$ .
$\bullet \bullet $ Every real finite-dimensional vector space is isomorphic to some $\mathbb R^n$ .
$\bullet \bullet \bullet $ Every linear map can be represented by a matrix if you brutally and non-canonically choose bases at the source and at the target.
$\bullet \bullet \bullet \bullet$ Every metric space is isometric to a subspace of a normed vector space .
$\bullet \bullet \bullet \bullet \bullet$ Every Stein manifold is isomorphic to a holomorphic submanifold of some $\mathbb C^n$
$\bullet \bullet \bullet \bullet \bullet \bullet \cdots$

And yet you should most of the time forget about these results and study groups, vector spaces, linear maps, ... intrinsically just as your teachers and all mathematicians do.
These pseudo-reductions to substructures of standard structures in general lead nowhere.
Results like Whitney's embedding theorem should essentially be enjoyed from an aesthetic point of view, but they are certainly not a panacea .

Answer 2

I agree with you that most books on differential geometry, if not all, start with manifold theory and don't even mention Whitney's embedding theorems (weak and/or strong). As pointed out in previous comments, this is because you want a more powerful and concrete machinery to work with.

However, since you are studying General Relativity and many interesting results in that field involve differential topology (e.g., Hawking-Penrose singularity theorems), I wanted to refer to a notable exception to that abstract manifold approach: Guillemin & Pollack's book on Differential Topology. This is what they explicitly say in the Preface:

"The book is divided into four chapters. Chapter l contains the elementary theory of manifolds and smooth mappings. We define manifolds as subsets of Euclidean space. This has the advantage that manifolds appear as objects already familiar to the student who has studied calculus in $\mathbf{R}^2$ and $\mathbf{R}^3$; they are simply curves and surfaces generalized to higher dimensions. We also avoid confusing the student at the start with the abstract paraphernalia of charts and atlases."

So, I would say that for a physicist studying general relativity, many ideas can be fully understood in the context of submanifolds of Euclidean spaces without having to learn all the theory of manifolds.

In fact, being completely honest, I know many string theorists who don't even know what a topological space is! Their mathematical knowledge of general relativity is more or less the same Einstein had a hundred years ago (Landau & Lifschitz's book or Weinberg's).

Hope it helps. Oswaldo.