  This conference talk elaborates on a recently discovered mapping procedure by
which classical orbits and path integrals for the motion of a point particle in
flat space can be transformed correctly into those in curved space. This
procedure evolved from well established methods in the theory of plastic
deformations where crystals with defects are described mathematically by
applying nonholonomic coordinate transformations to ideal crystals. In the
context of time-sliced path integrals, there seems to exists a quantum
equivalence principle which determines the measures of fluctating orbits in
spaces with curvature and torsion. The nonholonomic transformations produce a
nontrivial Jacobian in the path measure which in a curved space produces
  an additional term proportional to the curvature scalar canceling a similar
term found earlier by DeWitt from a naive formulation of Feynman's time-sliced
path integral. This cancelation is important in correctly describing
semiclassically and quantum mechanically various systems such as the hydrogen
atom, a particle on the surface of a sphere, and a spinning top. It is also
indispensible for the process of bosonization, by which Fermi particles are
redescribed in terms of Bose fields.
