A local Blaschke-Petkantschin formula in a Riemannian manifold

TL;DR

This paper derives a local Blaschke-Petkantschin formula for Riemannian manifolds, expressing how the Jacobian of a specific parametrization relates to curvature, with explicit expansions as the radius approaches zero.

Contribution

It introduces a novel local formula for Riemannian manifolds that accounts for curvature corrections in the Jacobian of point configurations.

Findings

Derived the Jacobian determinant for point configurations in Riemannian manifolds.

Provided an explicit expansion involving Ricci curvature as the radius tends to zero.

Extended the flat case formula to include local curvature effects.

Abstract

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.