Path integral approach for spaces of nonconstant curvature in three dimensions

In this contribution, I show that it is possible to construct three-dimensional spaces of nonconstant curvature, i.e., three-dimensional Darboux spaces. Two-dimensional Darboux spaces have been introduced by Kalnins et al., with a path integral approach by the present author. In comparison to two dimensions, in three dimensions it is necessary to add a curvature term in the Lagrangian in order that the quantum motion can be properly defined. Once this is done, it turns out that, in the two three-dimensional Darboux spaces which are discussed in this paper, the quantum motion is similar to the two-dimensional case. In D3d-I, we find seven coordinate systems which separate the Schroedinger equation. For the second space, D3d-II, all coordinate systems of flat three-dimensional Euclidean space which separate the Schroedinger equation also separate the Schroedinger equation in D3d-II. I solve the path integral on D3d-I in the (u, v, w) system and on D3d-II in the (u, v, w) system and in spherical coordinates