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Related question: embedding of $\mathbb{RP}^2$ in $\mathbb{R}^4$

I want to show $F[x,y,z]=(x^2−y^2,xy,xz,yz)$ is an immersion.

I know how to do it in local charts, for example by computing the jacobian of the composite map $(x,y) \mapsto [x,y,z]\mapsto (x^2−y^2,xy,xz,yz)$. (plus other charts)

But is it possible by examining the push-forward $F_*:T_PM \to T_{F(P)}N$?

What I tried is that if I first look at a curve in $S^2$ given by $x(t)^2+y(t)^2+z(t)^2=1$, then $xx'+yy'+zz'=0$ where $x=x(0),\ x'=x'(0)$ (similarly for $y,z$) Then computing $\dfrac d {dt}F(x(t),y(t),z(t))$ at $t=0$ gives $(2xx'-2yy',x'y+xy',x'z+xz',y'z+yz')$. So if I consider case $z\neq0$, I can separate it to $x'v+y'w$ where $v,w \in \mathbb{R}^4$. So I can finish this by showing $v,w$ is linearly independent. (also completing cases $y,z \neq 0$)

  1. Is it right? and how can I then conclude for a curve in $\mathbb{RP}^2$?
  2. Is there any method when I consider tangent vectors as derivations?
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1 Answer 1

Note that $ F(p)=F(-p)$ so that $F$ is well-defined on $P^2\mathbb{R}$. So we have a claim : $dF$ has full rank on $S^2$. That is, since $P^2\mathbb{R}$ is compact so it is an embedding.

Proof : $$ dF = \left( \begin{array}{ccc} 2x & -2y & 0 \\ y & x & 0 \\ z & 0 & x \\ 0 & z & y \end{array} \right) $$

In $dF$, consider minor $3\times 3$ matrix $A$ whose determinant is $-2xyz$.

If $xyz\neq 0$, then $dF$ has rank $3$. Rank is a continuous function on $S^2$ so that $dF$ has rank $2$ on $S^2$.

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