  We give here a covariant definition of the path integral formalism for the
Lagrangian, which leaves a freedom to choose anyone of many possible quantum
systems that correspond to the same classical limit without adding new
potential terms nor searching for a strange measure, but using as a framework
the geometry of the spaces considered. We focus our attention on the set of
paths used to join succesive points in the discretization if the time-slicing
definition is used to calculate the integral.If this set of paths is not
preserved when performing a point transformation, the integral may change. The
reasons for this are geometrically explained. Explicit calculation of the
Kernel in polar coordinates is made, yielding the same system as in Cartesian
coordinates.
