# On the numerical computation of Killing and conformally Killing vector fields on compact Riemannian manifolds

## TL;DR

This paper introduces a finite element-based numerical method to compute Killing and conformal Killing vector fields on compact Riemannian manifolds by transforming the problem into a symmetric eigenvalue problem.

## Contribution

It presents a novel approach that reduces the overdetermined PDE systems to symmetric eigenvalue problems, applicable in any dimension and manifold type.

## Key findings

- Method successfully computes vector fields on 2D manifolds
- Validates approach with numerical results in two dimensions
- Applicable to arbitrary compact Riemannian manifolds

## Abstract

The defining equations for Killing vector fields and conformal Killing vector fields are overdetermined systems of PDE. This makes it difficult to solve the systems numerically. We propose an approach which reduces the computation to the solution of a symmetric eigenvalue problem. The eigenvalue problem is then solved by finite element techniques. The formulation itself is valid in any dimension and for arbitrary compact Riemannian manifolds. The numerical results which validate the method are given in two dimensional case.