explicit formula for embedding projective spaces into euclidean space

Question

i can't seem to find very many good answers for this. most of the theorems out there use cohomology methods (Stiefel-Whitney classes, etc.) for proving that projective spaces can be embedded into Euclidean spaces of certain dimensions. Don Davis has a good table of minimal embedding dimensions. this paper (https://www.sciencedirect.com/science/article/pii/S0926224508000661) has some explicit formulas for isometric embeddings of projective spaces into Euclidean spaces, but the dimension of the target gets pretty large. for example the minimal dimension for an isometric embedding of RP^5 is 20, but it can be embedded in R^9. the question is: where can i find an explicit formula for such an embedding? or is there at least some procedure for me to figure one out? or am i just stuck running through the Whitney embedding theorem to try to get explicit maps?

Answer 1

Not sure if you are asking for any concrete embedding to $\Bbb{P^n(R)}\to \Bbb{R}^{N_n}$ or for one with $N_n$ small (if so look at this post). Anyway it is worth mentioning that for $N_n=n+1+n(n+1)/2$ it works and the embedding has a particularly simple expression.

$\Bbb{P^n(R)}$ is homeomorphic to $S^n/\pm 1$ through $$[p_0:\ldots:p_n] \to \pm (\frac{p_0}{\|p\|},\ldots,\frac{p_n}{\|p\|}), \qquad \|p\|=\sqrt{\sum_j p_j^2}$$

$S^n/\pm 1$ is a variety with coordinate ring $\Bbb{R}[S^n]^{\pm 1}$, the subring of $$\Bbb{R}[S^n]=\Bbb{R}[x_0,\ldots,x_n]/(\sum_{j=0}^n x_j^2-1)$$ fixed by $x\to -x$.

$$\Bbb{R}[S^n]^{\pm 1}= \Bbb{R}[ \{ \prod_{j=0}^n x_j^{e_j},2| \sum_j e_j \}]/I=\Bbb{R}[\{x_ix_j\}_{i\le j}]/I$$ (where $/I$ is to recall those are polynomial rings quotiented by the ideal of functions vanishing on $S^n$)

That's it we have found an embedding $\Bbb{P^n(R)}\to S^n/\pm 1\to \Bbb{R}^{n+1+n(n+1)/2}$ which is $$[p_0:\ldots:p_n] \to \pm (\frac{p_0}{\|p\|},\ldots,\frac{p_n}{\|p\|})\to (\frac{p_0p_0}{\|p\|^2},\ldots,\frac{p_ip_j}{\|p\|^2},\ldots,\frac{p_np_n}{\|p\|^2})$$