On geometrical representation of the Jacobian in a path integral reduction problem

TL;DR

This paper derives a geometric expression for the Jacobian in path integral reduction, relating it to scalar curvatures and geometric quantities of a Riemannian manifold with symmetry group action.

Contribution

It provides a new geometric formula for the Jacobian in path integral reduction involving scalar curvatures and curvature-related terms.

Findings

Expresses the Jacobian as a difference of scalar curvatures.

Links the Jacobian to the second fundamental form and principal bundle curvature.

Provides a geometric interpretation of the Jacobian in the reduction problem.

Abstract

The geometrical representation of the Jacobian in the path integral reduction problem which describes a motion of the scalar particle on a smooth compact Riemannian manifold with the given free isometric action of the compact semisimple Lie group is obtained. By using the formula for the scalar curvature of the manifold with the Kaluza--Klein metric, we present the Jacobian as difference of the scalar curvature of the total space of the principal fibre bundle and the terms that are the scalar curvature of the orbit space, the scalar curvature of the orbit, the second fundamental form of the orbit and the square of the principle fibre bundle curvature.