Ricci Curvature as a Partial Differential Equation

Abstract

Throughout this chapter, we shall investigate the consequences of viewing the expression for the Ricci tensor in terms of the metric as a partial differential operator. In other words, given a metric g, its Ricci curvature

$$ Ric\left( g \right) = r $$

is computed locally in terms of the first and second partial derivatives of g. We will think of r as prescribed and wish to investigate the properties of the metric. Some natural questions that arise are:

• (i) Given a Ricci candidate r, is there a metric g satisfying (5.1)?

• (ii) What conditions on r (on a compact manifold, say) insure uniqueness of such a metric (up to a constant multiple)?

• (iii) How does the smoothness of r influence the smoothness of g?