Dimension of the isometry group in three-dimensional Riemannian spaces

TL;DR

This paper establishes intrinsic, explicit, and algorithmic conditions for three-dimensional Riemannian metrics to admit isometry groups of various dimensions, enhancing the classification of symmetries in geometric spaces.

Contribution

It provides new intrinsic and explicit criteria for the existence of isometry groups in 3D Riemannian spaces, improving upon previous invariant-based methods.

Findings

Derived necessary and sufficient conditions for isometry group dimensions

Provided an intrinsic and explicit labeling method

Enhanced symmetry classification in 3D Riemannian geometry

Abstract

The necessary and sufficient conditions for a three-dimensional Riemannian metric to admit a group of isometries of dimension $r$ acting on s-dimensional orbits are obtained. These conditions are Intrinsic, Deductive, Explicit and ALgorithmic and they offer an IDEAL labeling that improves previously known invariant studies.