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Let $F:\mathbb{S}^{3}\to \mathbb{S}^{2}$, where $F(x,y,u,v)=(2.(xu+yv),2.(xv-yu),,u^{2}+v^{2}-x^{2}-y^{2})$. A submersion has a rank equal to dimension of codomain, then the work is prove that $rank F=2$. In my notes the author prove that there exists a minor with ordem $3$ with determinant non-zero, but i don't understand this, because in my definition i should prove that there exist a minor with order two with non-zero determinant and every minor with order $3$ has determinant zero. Lastly, there exists a way to prove that $F$ is a submersion without calculate the Jacobian?

Cézar Bezerra
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The author probably proves that there is a minor of order $3$ with non-zero determinant of the derivative of the map $$\widetilde{F}:\mathbb{R}^4 \to \mathbb{R}^3$$ (on points of the sphere, of course), which is just your map $F$ not restricted to the spheres. Note that this is a map that goes from a space of dimension $4$ to one of dimension $3$. This will imply that the derivative of the restricted map to the spheres is surjective by linear algebra arguments only.

There are other definitions of the Hopf fibration which may make it "easier/less laborious" to prove that it is a submersion (see here for examples). In your case, with such an explicit formula, I think this is the most straightforward way.

Aloizio Macedo
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    How can you prove by linear algebra arguments only that the restriction is surjective? – Watanabe Dec 28 '19 at 16:00
  • @Watanabe the differential maps $\mathbb{R}^4$ to $\mathbb{R}^3$(viewed as vector spaces) and its linear. – user57 Dec 10 '23 at 10:29