A General Form of the Constraints in the Path Integral Formula

A form of the constraints, specifying a $D$-dimensional manifold embedded in $D+1$ dimensional Euclidean space, is discussed in the path integral formula given by a time discretization. Although the mid-point prescription is privileged in the sphere $S^D$ case, it is more involved in generic cases. An interpretation on the validity of the formula is put in terms of the operator formalism. Operators from this path integral formula are also discussed.