How to construct a embedding from $\mathbb{R}{\text P}^{2n+1}$ to $\mathbb{R}^{4n+1}$

Question

Whitney's embedding theorem states that any smooth $n$-manifold $M$ can be smoothly embedded into $\Bbb R^{2n}$. And we know $\mathbb{R} {\text P}^{2n+1} $ is a $2n+1$-manifold, it can be embedded into $\mathbb R^{4n+2}$ according to the theorem. But I want to know if it can be embedded into $\mathbb R^{4n+1}$, is there any method to construct the embedding?

Answer 1

For each $n > 0$, $\mathbb{R}P^{2n+1}$ embeds into $\mathbb{R}^{4n+1}$. For $n=1$, this is a result of Wall that all $3$-manifolds embed into $\mathbb{R}^5$:

Wall, C. T. C., All 3-manifolds imbed in 5-space, Bull. Am. Math. Soc. 71, 564-567 (1965). ZBL0135.41603.

Further, in

Thomas, Emery, Embedding manifolds in Euclidean space, Osaka J. Math. 13, 163-186 (1976). ZBL0328.57009.

Thomas states that by combining results of Haefliger, Hirsch, Massey, and Petersen, it follows that all orientable $m$-manifolds embed into $\mathbb{R}^{2m-1}$, provided $m> 4$. Of course, $\dim \mathbb{R}P^{2n+1} = 2n+1 > 4$ when $n\geq 2$, and $\mathbb{R}P^{2n+1}$ is orientable. So these results cover the case of emedding $\mathbb{R}P^{2n+1}$ into $\mathbb{R}^{4n+1}$.

I do not know of any explicit embeddings.