Analysis of the electronic density using Riemannian geometry

Local-scaling transformations induce density-dependent deformations of the geometry, i.e., new metrics. We derived the Riemannian geometry tensors for the transformed metrics. We evaluated numerically the curvature scalar for atoms, and it reveals the shell structure. The change of sign of the curvature scalar from negative to positive corresponds to a maximum in the radial density, and the opposite change of sign to a minimum. We also explored some density-dependent conformally flat metrics. In special cases, we obtained measures of the curvature of the electronic density proportional to the Laplacian of the density and to the functional derivative of von Weizsäcker's kinetic energy functional or Bohm's quantum potential.