Condider $f:\mathbb{RP}^2 \rightarrow \mathbb{RP}^5$ given by
$$f[x,y,z]=[x^2,y^2,z^2,yz,xz,xy]$$
I would like to show that $f$ is a smooth embedding, meaning that it is a smooth immersion and a topological embedding. For $f$ to be a smooth immersion, I need the differential to be injective. This is given by
$$df=\begin{pmatrix}
2x & 0 & 0\\
0 & 2y & 0\\
0 & 0 & 2z\\
0 & z & y\\
z & 0 & x\\
y & x & 0\\
\end{pmatrix}.$$
However, I am not sure how homogeneous coordinates influence this. Do I need to use charts rather?
Showing map $\mathbb{RP}^2 \rightarrow \mathbb{RP}^5$ is smooth embedding
differential-geometrymanifoldssmooth-manifolds
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Best Answer
Yes, you need to use charts. What you have written is not $df$, it is the differential of a map $\mathbb{R}^3 \to \mathbb{R}^6$. But we can restrict $f$ to an affine chart $$ f[x,y,1] = [x^2,y^2,1,y,x,xy] $$ and compute its differential as a map $\mathbb{R}^2 \to \mathbb{R}^5$ $$ df = \begin{bmatrix} 2x & 0\\ 0 & 2y\\ 0 & 1\\ 1 & 0\\ y & x \end{bmatrix} $$
Edit: Here's how to finish the argument. By computing the differential you can show that $f$ is a smooth immersion. Next we show that $f$ is injective. Suppose that $f[x,y,z] = f[a,b,c]$. Then $x^2=a^2,y^2=b^2,$ and $z^2=c^2$ (since e.g. $x^2=-a^2$ is impossible) so $x=\pm a, y=\pm b, z=\pm c$. Now use the equations $xy=ab,yz=bc,xz=ac$ to see that either $x=a,y=b,$ and $z=c$ or $x=-a, y=-b$, and $z=-c$. In either case $[x,y,z]=[a,b,c]$ so $f$ is injective.
Now $f$ is an injective smooth immersion defined on compact manifold, so it is an embedding.