Is the Inverse Jacobian used to Transform Flat Space into Curved Space?

[friend]

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.

 

[friend]

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.

On searching the Web, I found that the inverse Jacobian is used to find robot joint angles with respect to each other, given an (x,y,z) componets. For each set of angles of a joint in a robotic arm there is a unique corresponding (x.y.z) coordinates. But the opposite is not necessarily true. If there are too many joints, then there may be many ways to produce an (x,y,z) coordinate. But I think it is true that if dimension of the curved space that the inverse jacobian is used to transform flat space into is of the same dimension as the flat space, then there is always a one-to-one correspondence. Is this true? Thanks.