I can do all the questions apart from part iv). After some work, I got . Unfortunately, the image of this under the metric is not 1/im(z)^2 as it ought to be.
Ok I've had another go. Can you tell me where I go wrong. Think of R^2 as C and let z=x+iy. Then, a(t)=z+t is a generating curve for e=(1,0).
so whose derivative is so that evaluating at t=0 gives which appears even further from the truth!. I think it hinges on what the derivative of is. My reasoning was which differentaing w.r.t t gives . Thanks