Emedding of $\mathbb{ RP}^3$

Question

Is there a simple formula for an embedding (homeomorphic onto its image) of $\mathbb{RP}^3$ in some Euclidean space? I have seen a simple formula for $\mathbb{RP}^2$ in $\mathbb R^4$, but I can't find much of anything on $\mathbb{RP}^3$. I am not asking for an immersion, but a true embedding.

Answer 1

Thinking of $\mathbb{RP}^n$ as $S^n$ mod antipode, and thinking of $S^n$ as the unit $n$-sphere $\{ x_0 + \dots + x_n^2 = 1 \} \subsetneq \mathbb{R}^{n+1}$, we can consider the map

$$S^n \ni (x_0, \dots x_n) \mapsto (x_0^2, \dots x_{n-1}^2, x_0 x_1, x_1 x_2, \dots x_{n-1} x_n) \in \mathbb{R}^{2n}.$$

This is a slight variant of a Veronese embedding. It factors through $\mathbb{RP}^n$ and produces an embedding (at least topologically; I haven't checked whether it's an immersion). Note that $x_n^2$ is omitted since it's determined by $x_0^2, \dots x_{n-1}^2$. The first $n$ coordinates produce a map which identifies two points of $S^n$ if each of their coordinates differs by a possibly different sign, and the remaining $n$ coordinates are there to ensure that all the signs match, so we only identify points which differ by a single global sign.