Which smooth compact n-manifolds admit Riemannian metrics of constant Ricci curvature? A direct variational approach sheds some interesting light on this problem, but by no means answers it. This article surveys some recent results concerning both Einstein metrics and the associated variational problem, with the particular aim of highlighting the striking manner in which the 4-dimensional case ...

This paper is an introduction to Riemannian geometry, with an aim towards proving the Hopf-Rinow theorem on complete Riemannian manifolds. We assume knowledge of the basics of smooth manifolds, including the tangent and cotangent bundles and vector fields. After a brief introduction to tensors, we develop the foundations of Riemannian geometry: geodesics, the exponential map, and the Riemannian...

The local geometry of a Riemannian symmetric space is described completely by the Riemannian metric and the Riemannian curvature tensor of the space. In the present article I describe how to compute these tensors for any Riemannian symmetric space from its Satake diagram, in a way that is suited for the use with computer algebra systems; an example implementation for Maple Version 10 can be fou...

In this paper we consider the space of those probability distributions which maximize the q-Rényi entropy. These distributions have the same parameter space for every q, and in the q = 1 case these are the normal distributions. Some methods to endow this parameter space with Riemannian metric is presented: the second derivative of the q-Rényi entropy, Tsallis-entropy and the relative entropy gi...