I am working on the following excercise:
Let $F:R^3\to R^6$ be given by $F(x,y,z)=(x^2,y^2,z^2,\sqrt{2}xy,\sqrt{2}xz,\sqrt{2}yz)$.
i) Show that $M=F(R^3-\{0\})$ is a submanifold of $R^6$.
ii) Show that the intersection of $M$ with $S^5$ is a submanifold of $S^5$ diffeomorphic to $RP^2$, called Veronese surface.
I have trouble finding which atlas I should equip the intersection with. I understand that it will be a 2-submanifold of $S^5$. My guess is that it would locally be the graph of function, but I coulnd't work that idea through the equation of the 5-sphere. Any ideas?