Prove that the Hopf map $\phi:S^3 \to S^2$ with $\phi(x,y)=(2x\bar y,|x|^2-|y|^2)$ is a submersion.
I need to show for that map rank $d\phi=2$. But how can I find partial derivative of $d(2x\bar y)/dy$. I'm stuck. Thank you.
Hopf map and submersion
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riemannian-geometry
1 Answers
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$$ K = x \bar{y} = (a + bi) (u - vi) = au + bv + (bu - av) i $$
$$ \partial K / \partial u = a + bi \\ \partial K / \partial v = b - ai \\ $$
It may be easiest to write out a $4 \times 4$ real determinant rather than a $2 \times 2$ complex one.