Normalizing a normally distributed vector to unit length If I have a random vector $\mathbf{y}$ generated from multivariate gaussian distribution $\mathcal{N}(\mathbf{0}, \mathbf{C})$, then I normalized it to unit length, which is,
$$\mathbf{y} \sim \mathcal{N}(\mathbf{0}, \mathbf{C}),$$
$$ \mathbf{z} = \frac{\mathbf{y}}{||\mathbf{y}||}.$$ 
Is it possible to determine the type of the distribution of $\mathbf{z}$? 
If $\mathbf{y} \sim \mathcal{N}(\mathbf{0},\mathbf{I})$, I think $\mathbf{z}$ should satisfy a multivariate uniform distribution with each of its components is identically and independently distributed from $\mathcal{U}(-1, 1)$. Am I correct? and how to make a proof?

No, if $y\sim N(0,I)$, then $z$ is uniformly distributed on the unit sphere. So the components can not be independent.
