Hard Whitney embedding theorem

This article is about an embedding theorem, viz about sufficient conditions for a given manifold (with some additional structure) to be realized as an embedded submanifold of a standard space (real or complex projective or affine space)
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Statement

A compact connected differential manifold of dimension $n$ can be embedded inside $\mathbb {R} ^{2n}$.

This is an improvement on the Whitney embedding theorem, which states that any compact connected differential manifold of dimension $n$ can be embedded inside $\mathbb {R} ^{2n+1}$ and immersed inside $\mathbb {R} ^{2n}$.