The Geometry and Topology of 3-Manifolds and Gravity

TL;DR

This paper explores a novel parameterization of 3-D Riemannian structures using gauge transformations, extending concepts from 2-D geometry and relating to 3-D gravity coupled with topological matter.

Contribution

It introduces a new finite-dimensional gauge transformation framework for parameterizing 3-D Riemannian geometries, generalizing 2-D conformal methods.

Findings

A finite-dimensional gauge group replaces conformal transformations in 3-D.

Parameterization connects 3-D geometry with topological matter in gravity.

Provides a new perspective on 3-D geometric structures in gravitational theories.

Abstract

It is well known that one can parameterize 2-D Riemannian structures by conformal transformations and diffeomorphisms of fiducial constant curvature geometries; and that this construction has a natural setting in general relativity theory in 2-D. I will show that a similar parameterization exists for 3-D Riemannian structures, with the conformal transformations and diffeomorphisms of the 2-D case replaced by a finite dimensional group of gauge transformations. This parameterization emerges from the theory of 3-D gravity coupled to topological matter.