Is there any study of the problems associated with the use of the inverse jacobian to go from flat space(time) to curved space(time)? I know they use the jacobian in curvilinear coordinates that parameterize flat space to convert the volume element in curved spaces to volume elements in flat spaces. And I think the jacobian is used to go from spaces with curvature and torsion to flat spaces. This can be done because all spaces are locally flat, right? So is there any use of the inverse jacobian to express things in flat space to things in curved space? Thanks.