Hello, I have geometry questions.

I need your help.

Thanks a lot.

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. Problem 6: The product manifold N x M admits two projection maps, which we denote TN : N XM - N TM : N XM - M. Prove that the sum of the induced maps of tangent bundles TRNOTTM : T(N X M) - TN x TM is a diffeomorphism. Problem 7 Argue that 2 x 2 matrices of rank 1 is a manifold. Denote this space by M1 (2, 2), i.e. we consider M1 (2, 2) to be a subspace of M2,2 (R) = 4