So I'm pretty new to studying manifolds and have little to no background on differential geometry, but this is a question from lecture notes on a multivariable analysis unit:
Show that $S:=\{(x^2,y^2,z^2,yz,xz,xy)|x,y,z \in \mathbb R, x^2+y^2+z^2=1\}$ is a smooth 2-submanifold of $\mathbb R^6$ and show that the projection of $S$ onto the last three coordinates (The Roman Surface $R:=\{(yz,xz,xy)|x,y,z \in \mathbb R, x^2+y^2+z^2=1\}$) is not a smooth 2-submanifold of $\mathbb R^3$
For the first part I tried to construct an atlas for $S$, given by:
$\phi_1: D \to S$, where D is the open unit disc around the origin in $\mathbb R^2$, such that $$(x,y) \mapsto (x^2,y^2,1-x^2-y^2,y\sqrt{1-x^2-y^2},x\sqrt{1-x^2-y^2},xy)$$ and $\phi_2$ and $\phi_3$ are defined similarly by isolating $x$ and $y$, respectively, on $x^2+y^2+z^2=1$
Now what I would like to know is:
1) Is this enough to show that $S$ is a smooth 2-submanifold of $\mathbb R^6$?
2) If the answer to 1) is yes, then, is this generally the best way to prove that a subset is a submanifold?
3) For the second part I have no idea how to show something is not a submanifold and would appreciate any help on the matter
