In a research problem about detection theory I faced with the following question. How can I find the conditional Gaussian probability density function
$$ f({\mathbf w} \; | \; \|{\mathbf w}\|^2)$$ where $\mathbf w$ is an $N$ dimensional random vector with Gaussian distribution with Zero mean and co-variance matrix $\mathbf I$ ($N\times N$ identity matrix) and also $\| {\mathbf w}\| $ is the 2-Norm of vector $\mathbf w$?.
HINT
In two dimensions, it is easy to see that $\|{\mathbf w}\|^2=w$ is a circle of radius $w$. considering that the Gaussian distribution with the given $\mathbf I$ is rotationally symmetrical, we may say that the conditional distribution given is uniform over the circle centered at the origin (the means are $0$s).
Generalizing this intuition, I suspect, that the conditional distribution in question is uniform over the surface of the suitable hyper sphere.
EDIT
The distribution is uniform over the $(n-1)$-sphere in the $n$ dimensional Euclidean space. The surface of the $(n-1)$-sphere centered at the origin is described by
$$\sum_{i=1}^n x_i^2=w^2.$$
The surface area of the same is
$$V_n(w)=\frac{\pi^{\frac n2}}{\Gamma\left(\frac n2+ 1\right)}w^n.$$
So, the distribution is $$\frac1{V_n(w)} \text{ if } \sum_{i=1}^n x_i^2=w^2, 0\text{ otherwise .}$$
See also Marsaglia's algorithm.