embedding projective plane in 4-space?

Question

Is it possible to embed projective plane in 4-space? If not what is the reason and what is the smallest singularity set?

Answer 1

Let $f:\mathbb{S}^2→\mathbb{R}^4$ be defined by $$f(x,y,z)=(yz,xz,xy,ax^2+by^2+cz^2)$$ this gives an embedding of $\mathbb{RP}^2$ in $\mathbb{R}^4$.

We have $f(x,y,z)=f(−x,−y,−z)$, so we get a map $F:\mathbb{RP}^2→\mathbb{R}^4$. The differential $dF$ has rank 2, so this is an immersion and $f$ is injective. Then have to embedding.