Isometric and metamorphic operations on the space of local fundamental   measures
  We consider symmetry operations on the four-dimensional vector space that is spanned by the local versions of the Minkowski functionals (or fundamental measures): volume, surface, integral mean curvature, and Euler characteristic, of an underlying three-dimensional geometry. A bilinear combination of the measures is used as a (pseudo) metric with ++-- signature, represented by a 4x4 matrix with unit entries on the counter diagonal. Six different types of linear automorphisms are shown to leave the metric invariant. Their generators form a Lie algebra that can be grouped into two mutually commuting triples with non-trivial structure constants. We supplement these six isometric operations by further ten transformations that have a metamorphic (altering) effect on the underlying geometry. When grouped together, four different linear combinations of the metamorphic generators form a previously obtained third-rank tensor. This is shown to describe four different types of mutually commuting "shifting" operations in fundamental measure space. The relevance for fundamental measures density functional theory is discussed briefly. 
Matthias Schmidt
math-ph cond-mat.other math.MP