Symmetries in Differential Geometry: A Computational Approach to Prolongations

TL;DR

This paper develops a systematic computational method to analyze overdetermined systems from conformal Killing tensors, enabling the discovery of symmetries and higher-order operators in differential geometry.

Contribution

It introduces a new general method for closing overdetermined systems for any rank of conformal Killing tensors, applicable to various Killing equations and conditions.

Findings

Identifies higher symmetry operators for the Laplace operator.

Finds first-order symmetry operators for the Yamabe case.

Proposes conjectures on second-order symmetries of the Yamabe operator.

Abstract

The aim of this work is to develop a systematic manner to close overdetermined systems arising from conformal Killing tensors (CKT). The research performs this action for 1-tensor and 2-tensors. This research makes it possible to develop a new general method for any rank of CKT. This method can also be applied to other types of Killing equations, as well as to overdetermined systems constrained by some other conditions. The major methodological apparatus of the research is a decomposition of the section bundles where the covariant derivatives of the CKT land via generalized gradients. This decomposition generates a tree in which each row represents a higher derivative. After using the conformal Killing equation, just a few components (branches) survive, which means that most of them can be expressed in terms of lower order terms. This results in a finite number of independent jets. Thus, any higher covariant derivative can be written in terms of these jets. The findings of this work are significant methodologically and, more specifically, in the potential for the discovery of symmetries. First, this work has uncovered a new method that could be used to close overdetermined systems arising from conformal Killing tensors (CKT). Second, through an application of this method, this research finds higher symmetry operators of first and second degree, which are known by other means, for the Laplace operator. The findings also reveal the first order symmetry operators for the Yamabe case. Moreover, the research leads to conjectures about the second order symmetries of the Yamabe operator.