Path Integral Approach for Spaces of Non-constant Curvature in Three Dimensions

In this contribution I show that it is possible to construct three-dimensional spaces of non-constant 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 $\threedDI$ we find seven coordinate systems which separate the Schr\"odinger equation. For the second space, $\threedDII$, all coordinate systems of flat three-dimensional Euclidean space which separate the Schr\"odinger equation also separate the Schr\"odinger equation in $\threedDII$. I solve the path integral on $\threedDI$ in the $(u,v,w)$-system, and on $\threedDII$ in the $(u,v,w)$-system and in spherical coordinates.