Image of Sphere is a submanifold of $R^6$

Question

I was trying to prove that $\alpha(S^2)$ is a submanifold of $\mathbb{R}^6$.

$\alpha(x,y,z) = (x^2, y^2, z^2, \sqrt2 xy, \sqrt2 yz, \sqrt2 zx)$

$\textbf{Attempt}$

I started by proving that $\alpha(S^2)$ is contained in $S^5$, then I noticed that $S^5$ is a submanifold of $\mathbb{R}^6$ in this point can i conclude that $\alpha(S^2)$ is also a submanifold of $\mathbb{R}^6$?

If my approach is wrong, I would love some hints!

Thank you.

Answer 1

Hint: Verify that $\alpha$ is an immersion and then look at Homeomorphism from $P^2\mathbb{R}$ onto the image of $\mathbb{S}^2$ through the Veronese map .