Triangulated Surfaces and Polyhedral Structures

Abstract

In this chapter we introduce the foundational material that will be used in our analysis of triangulated surfaces and of their quantum geometry. We start by recalling the relevant definitions from Piecewise-Linear (PL) geometry, for which we refer freely to Rourke and Sanderson (Introduction to piecewise-linear topology. Springer-Verlag, New York, 1982) , Thurston (Three-dimensional Geometry and Topology, Vol 1. Princeton University Press, Berkeley, 1997). After these introductory remarks we specialize to the case of Euclidean polyhedral surfaces whose geometrical and physical properties will be the subject of the first part of the book. The focus here is on results which are either new or not readily accessible in the standard repertoire. In particular we discuss from an original perspective the structure of the space of all polyhedral surfaces of a given genus and their stable degenerations. This is a rather delicate point which appears in many guises in quantum gravity, and string theory, and which is related to the role that Riemann moduli space plays in these theories. Not surprisingly, the Witten–Kontsevich model by Kontsevitch (Commun Math Phys 147:1–23, 1992) lurks in the background of our analysis, and some of the notions we introduce may well serve for illustrating, from a more elementary point of view, the often deceptive and very technical definitions that characterize this subject. In such a framework, and in the whole landscaping of the space of polyhedral surfaces an important role is played by the conical singularities associated with the Euclidean triangulation of a surface. We provide, in the final part of the chapter, a detailed analysis of the geometry of these singularities. Their relation with Riemann surfaces theory will be fully developed in Chap. 2.