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

TL;DR

The paper introduces a rigorous regularization scheme for defining invariant geometric functionals on manifolds with conical defects by treating $M_α$ as a limit of smooth spaces, enabling finite expressions for quantities like the Euler characteristic $χ$ and Hirzebruch signature $τ$. It extends the framework to higher-curvature theories, showing Lovelock actions remain finite with surface contributions localized on the defect, and applies the method to black hole entropy in higher-derivative gravity and to 2D quantum models via the Polyakov-Liouville action. The approach yields results that align with established methods (e.g., Gibbons-Hawking) while providing a clear separation between regular-point contributions and defect-induced surface terms. Overall, the work provides a mathematically solid foundation for Riemannian geometry on singular spaces and yields physically meaningful quantities in gravitational thermodynamics and quantum gravity contexts.

Abstract

A consistent approach to the description of integral coordinate invariant functionals of the metric on manifolds ${\cal M}_α$ with conical defects (or singularities) of the topology $C_α\timesΣ$ is developed. According to the proposed prescription ${\cal M}_α$ 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 ${\cal M}_α$ 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 ${\cal M}_α$ 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 ${\cal M}_α$, and gives the same results as in the other methods.