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Regarding the post:

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

I want to understand why $F$ is an immersion. Since $\mathbb{R}P^{2}$ is the quotient of $\mathbb{S}^{2}$ by identifying the antipodal points and we have it suffices to show that the map $f: \mathbb{S}^{2} \subset \mathbb{R}^{3} \rightarrow \mathbb{R}^{4}$ given by $f(x,y,z)=(x^{2}-y^{2},yz,xz,xy)$ is an immersion right? because we have that $F \circ \pi= f$ where $\pi: \mathbb{S}^{2} \rightarrow \mathbb{R}P^{2} = \mathbb{S}^{2}$/~ is the projection map.

OK so I compute the Jacobian and get:

$\begin{bmatrix} 2x & -2y & 0 \\\ 0 & z &y \\ z & 0 & x \\ y & 0 & 0 \end{bmatrix}$

Would it suffice then to show this matrix has rank $3$? If this is not correct can you please explain why $F$ is an immersion? or any other approach is appreciated.

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No, that does not suffice. First of all, your maps natural domain is $\mathbb R^3$ and you are restricting it to $S^2$. Once you've restricted to $S^2$ it has a derivative by not a Jacobian. What you need to show is that non-zero tangent vectors are sent to non-zero vectors. So you have to argue that if $(a,b,c)\cdot (x,y,z)=0$ and if $(a,b,c)$ is not zero, then the product of $(a,b,c)$ with your matrix is non-zero. – Ryan Budney Oct 7 '11 at 2:43
up vote 3 down vote accepted

Let $f:P^2\to\mathbb R^4$ be your map and let $\pi:S^2\to P^2$ be the quotient map.

In order to check that $f$ is an immersion, we need only show that, for each point $p\in P^2$ there is an open neighborhood $U$ of $p$ in $P^2$ such that the restriction $f|_U:U\to\mathbb R^4$ is an immersion.

So fix $p\in P^2$. Let $q\in S^2$ be one of its preimages, and let $V\subseteq S^2$ be the open hemisphere of $S^2$ which has $q$ in its center. Let $U=\pi(V)$ which is an open neighborhood of $p$. The map $\pi|_V:V\to U$ is a diffeomorphism. It follows that to check that $f|_U$ is an immersion, it is enough to check that $(f\circ\pi)|_V:V\to\mathbb R^4$ is an immersion.

I am going to suppose that $q=(0,0,1)$. Let $W=\{(x,y)\in\mathbb R^2:x^2+y^2<1\}$ and let $\phi:(x,y)\in W\mapsto(x,y,\sqrt{1-x^2-y^2})$; this is a regular parametrization of $V$. To check that $(f\circ\pi)|_V:V\to\mathbb R^4$ is an immersion it is enough to check that $(f\circ\pi\circ \phi)|:W\to\mathbb R^4$ is an immersion. This you can do by computing the Jacobian of this map and showing it has rank $2$ at each point of $W$.

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