Suppose I have an $n$-dimensional Gaussian random variable with mean $\mu\in \mathbb{R}^n$ and covariance matrix $\sigma^2I_n$, where $I_n$ is the identity matrix.
If I condition on the distribution lying on the unit sphere and if $|\mu|=1$, I am told that the resulting probability distribution is the von Mises-Fisher distribution.
My question is: if I project the Gaussian onto the sphere, what is the resulting distribution? Does it have a name?