Immersion between spheres.

Question

Let $F:\mathbb{S}^{2}\to \mathbb{S}^{5}$ be given by $F(x,y,z)=(x^{2},y^{2},z^{2},\sqrt{2}xy,\sqrt{2}yz,\sqrt{2}xz)$. Prove that is an immersion. I have a minor with order three with determinant equal to $xyz$ then $Jac F(x,y,z)$ has rank $3$ except when $x=0$ or $y=0$ or $z=0$, now i have a problem for to be continue.Any tips?

Answer 1

You're right that the rank of ${\rm Jac \ } F(x,y,z)$ (viewed as a linear map from $\mathbb R^3 $ to $\mathbb R^6$) is less than $3$ when $x = 0$, $y = 0$ or $z = 0$. But this is not a problem! Your real goal is to prove that ${\rm Jac \ } F(x,y,z)$ is injective on the space of tangent vectors for any point $\vec x\in \mathbb S^2$. Any such tangent space is only two-dimensional. So you could approach this by finding two basis vectors for the tangent space, and verify that the images of these two tangent vectors under ${\rm Jac \ } F(x,y,z)$ are linearly independent.

For example, let's look at the $z = 0$ case. A typical point on $\mathbb S^2$ with $z = 0$ looks like this: $$ \vec x = (\cos \phi, \sin \phi, 0).$$ The tangent space at this point is generated by the basis vectors, $$ \vec v_1 =(-\sin \phi, \cos \phi, 0), \ \ \ \vec v_2 = (0, 0, 1).$$ Your task is to show that the images ${\rm Jac }F(\vec x)(\vec v_1)$ and ${\rm Jac }F(\vec x)(\vec v_2)$ are linearly independent. Hopefully this should be too difficult to verify!