Suppose I want to generate points uniformly from a sphere (surface) of dimension $d$. A given solution is generate $d$ 1-dimensional gaussian points and then normalize the vector. Generating many points this way gives a distribution that is uniform over the surface of the sphere.
However " Note that once the vector is normalized, its coordinates are no longer statistically independent."
I didn't get the reasoning behind it. And a follow-up question is that how to generate iid points on a surface of sphere?