Riemannian Geometry and the Prescribed Ricci Curvature Problem
By Anna Kervison, The University of Queensland
Riemannian geometry is a branch of non-Euclidean geometry developed by Bernhard Riemann, used to describe curved space. In Riemannian geometry, a manifold is a topological space that is locally flat. This means that for any neighbourhood on the manifold there exists an invertible map from that neighbourhood to Rn. For example, circles are one-dimensional manifolds but a figure eight is not as it cannot be projected into the Euclidean plane at the intersection. Surfaces such as the sphere and the torus are examples of two-dimensional manifolds.
The shape of a manifold is defined by the Riemannian metric, which is a measure of the length of tangent vectors and curves in the manifold. It can be thought of as locally a matrix valued function. The Ricci curvature is one of the most significant geometric characteristics of a Riemannian metric. It provides a measure of the curvature of the manifold in much the same way the second derivative of a single valued function provides a measure of the curvature of a graph. Determining the Ricci curvature of a metric is difficult, as it is computed from a lengthy expression involving the partial derivatives of components of the metric up to order two. In fact, without additional simplifications, the formula for the Ricci curvature given by this definition is essentially unmanageable.
One of the simplest examples of a manifold is simply Euclidean space – Rn. This project focused on Riemannian metrics on Rn that are rotationally invariant – metrics that do not change under rotations of Rn. This special class of metrics have the property that they are conformal to a flat metric, which means that, in Euclidean space the angles between vectors with respect to these metrics are the same as the angles between vectors with respect to a metric with zero curvature. A consequence of this is that the formula for the Ricci curvature on this class of metrics simplifies significantly and the fundamental problem of recovering a metric from its Ricci curvature – known as the prescribed Ricci curvature problem – reduces to a single ordinary differential equation. This single ordinary differential equation can then be used to find a sufficient condition for the global existence of a solution to the prescribed Ricci curvature equation and to show the solution is unique.