  Let ${\pmb M}$, ${\pmb N}$ and ${\pmb K}$ be $d$-dimensional Riemann
manifolds. Assume that ${\bf A}:=(A_n)_{n\in{\Bbb N}}$ is a sequence of
Lebesgue measurable subsets of ${\pmb M}$ satisfying a necessary density
condition and ${\bf x}:=(x_n)_{n\in {\Bbb N}}$ is a sequence of independent
random variables which are distributed on ${\pmb K}$ according to a measure
which is not purely singular with respect to the Riemann volume. We give a
formula for the almost sure value of the Hausdorff dimension of random covering
sets ${\bf E}({\bf x},{\bf A}):=\limsup_{n\to\infty}A_n(x_n)\subset {\pmb N}$.
Here $A_n(x_n)$ is a diffeomorphic image of $A_n$ depending on $x_n$. We also
verify that the packing dimensions of ${\bf E}({\bf x},{\bf A})$ equal $d$
almost surely.
