Ricci tensor under conformal change of metric as an elementary obstruction to certain Einstein metrics

Abstract

Gregorio Ricci-Curbastro, (1853-1925) Bologna, was an Italian mathematician and a professor at the University of Padua from 1880 -1925. He was the rst to introduce the systematic theory of tensor analysis in 1887 with a a major contribution later by his student Tullio Levi-Civita. However, the roots of tensor analysis were laid by the work of German mathematician Bernhard Riemann in Di erential Geometry. The beginning of the twentieth century was the emergence of the study of the Ricci Tensor due to its major role in the mathematical formulation of the theory of general relativity of Albert Einstein. Also, as one of the important geometric features that have been used to prove several major theorems in di erential geometry and topology. Here I will focus on the idea of the conformal change of metrics and what is the necessary and su cient condition for a metric to be conformal to another metric on the same manifold, and how geometric objects like Riemannian and Ricci curvature are related under this change.