Abstract In this paper, we consider a Riemannian manifold $M$ and the Poisson–Voronoi tessellation generated by the union of a fixed point $x_0$ and a Poisson point process of intensity $\lambda $ on $M$. We obtain a two-term asymptotic expansion, when $\lambda $ goes to infinity, of the mean number of vertices of the Voronoi cell associated with $x_0$. The 1st term of the estimate is equal to the mean number of vertices in the Euclidean setting, while the 2nd term involves the scalar curvature of $M$ at $x_0$. This settles with the proper and rigorous frame the former 2D statement from [ 19] and extends it to higher dimension. The key tool for proving this result is a new change of variables formula of Blaschke–Petkantschin type in the Riemannian setting, which brings out the Ricci curvatures of the manifold.