Abstract
Hassler Whitney proved the first general embedding theorem in 1936. He showed, among other things, that any differentiable manifold could be properly embedded in a higher-dimensional Euclidean space. His fundamental tools included Lebesgue measure theory and the notion of a cut-off function. In fact, any differentiable mapping into a suitable-dimensional Euclidean space can be approximated by such an embedding.
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Notes
- 1.
Here we refer only to Lebesgue measure in \(\mathbf{R}^n\).
- 2.
In fact, \(K_n =\frac{\pi ^\frac{n}{2}}{\Gamma (\frac{n}{2}+1)}\).
- 3.
We have not been able to ascertain who first described such an example, but we recall that Weierstrass formulated in 1872 an example of a continuous function which is nowhere differentiable [230], so he or others at that time might have known such an example.
- 4.
It is easy to see that such a covering exists. Simply take any countable covering \(\{\tilde{U}_j\}\) of \(M\) with coordinate chart mappings
$$ \tilde{h}_j: \tilde{U}_j \rightarrow \mathbf{R}^n, $$and then consider the countable collection of balls \(\{B_{\mu j}\}\) in the open set
$$ \tilde{h}_j(U_j) \subset \mathbf{R}^n $$of rational radii and rational center points. The collection of open sets \(\{\tilde{h}_j(B_{\mu j})\}\) will provide a covering of \(M\) from which one can construct the desired locally finite covering.
- 5.
The lecture notes on differential topology by Milnor [154] have a very readable and simplified account of Whitney’s original arguments in [245].
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Wells, R.O. (2017). Differentiable Manifolds. In: Differential and Complex Geometry: Origins, Abstractions and Embeddings. Springer, Cham. https://doi.org/10.1007/978-3-319-58184-2_12
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DOI: https://doi.org/10.1007/978-3-319-58184-2_12
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