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For the Jacobian resulting from the previously considered problem of the pathintegral reduction in Wiener path integrals for a mechanical system withsymmetry describing the motion of two interacting scalar particles on amanifold that is the product of a smooth compact finite-dimensional Riemannianmanifold and a finite-dimensional vector space, a geometric representation isobtained. This representation follows from the formula for the scalar curvatureof the original manifold endowed by definition with a free isometric smoothaction of a compact semisimple Lie group. The derivation of this formula isperformed using adapted coordinates, which can be determined in the principalfiber bundle associated with the problem under the study. These coordinates aresimilar to those used in the standard approach to quantization of Yang-Millsfields interacting with scalar fields.

On the geometric representation of the path integral reduction Jacobian  for a mechanical system with symmetry given on a manifold that is a product  of the total space of the principal fiber bundle and the vector space