Hi, I have geometry questions.

I need your help.

Thank you so much.

Problem 4: We dene a function (C x R by the formula 05(20321) = (2205, IZOI2 - I21|2)- Argue that if (zo, zl) is a unit vector, then (z0, 21) is also a unit vector. Compute the derivative of d). Consider a : S3 > 82 to be the restriction of q') to the unit Spheres. Show that the rank of Do is always two, i.e. that a is a submersion. Moreover, show that a(zo, 21) 2 0(26, 21) if and only if (20, 21) = /\\(z(',, 21) for some unit complex number A. The map a is called the Hopf bmtion