Discrete conformal variations and scalar curvature on piecewise flat two- and three-dimensional manifolds. (English) Zbl 1243.52014

Glickenstein, David

J. Differ. Geom. 87, No. 2, 201-238 (2011).

The article is concerned with the geometry of low-dimensional piecewise flat manifolds, which are closed topological manifolds constructed by identifying the boundaries of Euclidean triangles or Euclidean tetrahedra, whose 1-simplices have specified lengths. They are discrete analogues of 2 and 3-dimensional Riemannian manifolds.
The author defines the edge curvature, Ricci curvature, polyhedral scalar curvature, etc., as well as the conformal variation of piecewise flat manifolds via variation of angles. The author calculates the first and second variation of the Einstein-Hilbert-Regge functional, studies its convexity, and proves rigidity theorems analogous to those in the smooth case.

Reviewer: Rafael Herrera (Guanajuato)

MSC:

52B70	Polyhedral manifolds
53C20	Global Riemannian geometry, including pinching
53C24	Rigidity results

Keywords:

piecewise flat manifold; conformal variation; Einstein-Hilbert-Regger functional; rigidity