Entanglement Entropy and Spatial Geometry
  The entanglement entropy in a quantum field theory between two regions of space has been shown in simple cases to be proportional to the volume of the hypersurface separating the regions. We prove that this is true for a free scalar field in an arbitrary geometry with purely spatial curvature and obtain a complete asymptotic expansion for the entropy. 
Micheal S. Berger and Roman V. Buniy