Three-dimensional metrics as deformations of a constant curvature metric

TL;DR

This paper demonstrates that any three-dimensional Riemannian metric can be locally derived from a constant curvature metric through a deformation along a single direction, highlighting its significance in geometry and physics.

Contribution

It introduces a method to generate any 3D Riemannian metric via deformation of a constant curvature metric along one direction, offering new insights into geometric structures.

Findings

Any 3D Riemannian metric can be locally obtained by deformation.

The deformation involves a single directional change.

The result has implications in both geometry and physics.

Abstract

Any three-dimensional Riemannian metric can be locally obtained by deforming a constant curvature metric along one direction. The general interest of this result, both in geometry and physics, and related open problems are stressed.