Study of Ricci Curvature Tensor Under Conformal Transformation

Abstract

This study discusses how the Ricci tensor would transform under conformal trans- formations. It specifically worked toward the development of the Ricci curvature

tensor transformation equation. The laws of transformation of the curvature

tensors, that is, the Ricci tensor, the scalar curvature, and the Riemannian cur- vature tensor, have been mentioned in earlier works. Still, their discussion has

been chiefly succinct, without a detailed presentation. It led them to neglect geometric terms, namely, the Christoffel symbols, conformal factor, and their derivatives. The existence of this gap inspired the research. The aim was to provide a more precise and stricter view of the behaviour of the Ricci tensor

under a conformal transformation. The task of the study was to derive the for- mula for the transformation of the Ricci tensor in terms of the initial tensor and

expressions of the conformal factor. It also looked at the transformation of the scalar curvature and Riemannian curvature tensor and how sensitive they were to conformal changes. Explicit computation and coordinate-free Analysis were done using tensor calculus and differential geometry tools. The results were compared using the different dimensions to show the properties and invariants depending on the dimensions. Conformally transformed Ricci curvature tensors study has critical roles in General Relativity (GR) to study the space-time curvature in the asymptotically flat space and interpret the Einstein field equations. Geometric Analysis. It is also central to investigate the Yamabe problem and the existence

of canonical metrics. This Analysis is essential in theoretical physics, particu- larly in conformal field theory (CFT), where it is possible to study how curvature

responds to length-changing yet angular-preserving transformations.