Isometric and metamorphic operations on the space of local fundamental measures

TL;DR

This paper explores symmetry and transformation operations on the space of local fundamental measures in three-dimensional geometry, identifying isometric automorphisms and metamorphic transformations with implications for density functional theory.

Contribution

It characterizes the automorphisms and transformations of the fundamental measures space, revealing a Lie algebra structure and new shifting operations relevant to density functional theory.

Findings

Identified six isometric automorphisms preserving the metric.

Discovered ten metamorphic transformations with geometric effects.

Linked transformations to a third-rank tensor describing shifting operations.

Abstract

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.