On the Description of the Riemannian Geometry in the Presence of Conical Defects

TL;DR

This paper develops a rigorous mathematical framework for describing Riemannian geometry on manifolds with conical defects, enabling precise computation of topological invariants and gravitational actions in singular spaces.

Contribution

It introduces a method to treat conical singularities as limits of smooth spaces, allowing for exact calculations of invariants and actions in such geometries.

Findings

Explicit formulas for Euler numbers and signatures with conical singularities

Lovelock gravity is well-defined on manifolds with conical defects

Black hole entropy calculations are consistent in singular geometries

Abstract

A consistent approach to the description of integral coordinate invariant functionals of the metric on manifolds ${\cal M}_{\alpha}$ with conical defects (or singularities) of the topology $C_{\alpha}\times\Sigma$ is developed. According to the proposed prescription ${\cal M}_{\alpha}$ are considered as limits of the converging sequences of smooth spaces. This enables one to give a strict mathematical meaning to a number of invariant integral quantities on ${\cal M}_{\alpha}$ and make use of them in applications. In particular, an explicit representation for the Euler numbers and Hirtzebruch signature in the presence of conical singularities is found. Also, higher dimensional Lovelock gravity on ${\cal M}_{\alpha}$ is shown to be well-defined and the gravitational action in this theory is evaluated. Other series of applications is related to computation of black hole entropy in the higher derivative gravity and in quantum 2-dimensional models. This is based on its direct statistical-mechanical derivation in the Gibbons-Hawking approach, generalized to the singular manifolds ${\cal M}_{\alpha}$, and gives the same results as in the other methods.