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A geometric representation is found for the previously obtained path integralreduction Jacobian in Wiener-type path integral when quantizing a modelmechanical system, which is used to describe the motion of two interactingscalar particles on a product manifold (a smooth compact finite-dimensionalRiemannian manifold and vector space) with a given free isometric action of acompact semisimple Lie group. The reduction Jacobian we are dealing with wasobtained for the case when, as in gauge theories, dependent coordinates areused to locally describe the reduced motion. As in our similar works, theresult is based on the scalar curvature formula for the original manifold whichis viewed as a total space of the principal fiber bundle. The calculation ofthe Christoffel symbols and scalar curvature was performed in a specialnonholonomic basis, also known as the horizontal lift basis.

On the geometric representation of the path integral reduction Jacobian  in the path integral for interacting systems: The case of dependent  coordinates in the description of reduced motion on the orbit space