Prolongation and Killing two-tensors

TL;DR

This paper introduces a systematic prolongation method for Killing two-tensors, especially in locally symmetric spaces, and explores their relationship with Killing fields on such manifolds.

Contribution

It provides a new prolongation procedure for Killing two-tensors and clarifies the quadratic mapping from Killing fields to two-tensors in symmetric spaces.

Findings

Developed a systematic prolongation procedure for Killing two-tensors.

Applied the method to understand the quadratic mapping in locally symmetric spaces.

Enhanced understanding of the structure of Killing tensors in symmetric geometries.

Abstract

We present a systematic prolongation procedure and its implementation for Killing two-tensors, especially in the locally symmetric case. We use the resulting machinery to elucidate the natural quadratic mapping from Killing fields to Killing two-tensors on irreducible locally symmetric spaces of compact type.