On Spaces with the Maximal Number of Conformal Killing Vectors

TL;DR

This paper proves that a space admits the maximal number of conformal Killing vectors if and only if it is conformally flat, establishing a necessary and sufficient condition for maximal CKV symmetry.

Contribution

It provides a rigorous proof that conformal flatness is both necessary and sufficient for maximal conformal Killing vectors in a space.

Findings

Maximal CKVs imply conformal flatness.

Conformal flatness guarantees maximal CKVs.

The equivalence is proven rigorously.

Abstract

It is natural to expect and simple to prove that every conformally flat space possess the maximal number of conformal Killing vector fields (CKVs). On the other hand, it is interesting to ask whether the converse is true. Is conformal flatness a necessary condition for the existence of the maximal number of CKVs? In this review article it is proven that the answer is yes, a space admits the maximal number of CKVs if, and only if, it is conformally flat.