A local Blaschke-Petkantschin formula in a Riemannian manifold

In this paper, we show a local Blaschke-Petkantschin formula for a Riemannian manifold. Namely, we compute the Jacobian determinant of the parametrization of $(n+1)$-tuples of the manifold by the center and the radius of their common circumscribed sphere as well as the $(n+1)$ directions characterizing the positions of the $n+1$ points on it. We deduce from it a more explicit two-term expansion when the radius tends to $0$. This formula contains a local correction with respect to the flat case which involves the Ricci curvatures in the $(n+1)$ directions.