For Poisson-Voronoi tessellations of $\mathbb R^2$, the expected number of vertices on the boundary of the typical cell is 6. Proofs of this can be found in section 9.3.4 of Stochastic Geometry and Its Applications (3rd Edition) by Stoyan, Mecke, and Kendall (in fact it works for more general point processes)
Let $\lambda_0$ denote the intensity of the vertices formed by the Voronoi cells. Let $\lambda_2$ denote the intensity of the Poisson point process. Let $\bar{n}_{02}$ denote the expected number of edges emanating from the typical vertex and $\bar{n}_{20}$ denote the expected number of vertices on the boundary of the typical cell. They establish the identity $\lambda_2\bar{n}_{20}=\lambda_0\bar{n}_{02}.$
My question is does this identity hold for "nice" manifolds (like surfaces of constant curvature or those with more restrive conditions)?
Thanks!