*Smooth manifolds* are generalisations of smooth curves and surfaces to higher dimensions. A *Riemannian metric* is a mathematical object defined on a smooth manifold which allows us to talk about distances, angles, and curvatures. A smooth manifold equipped with a Riemannian metric is called a *Riemannian manifold*. Arguably, the most famous examples of Riemannian manifolds are the *three model spaces*: *Euclidean space*, the *round sphere*, and *hyperbolic space.*

Any smooth manifold admits infinitely many Riemannian metrics. Thus, mathematicians have been interested in the following question: *on a given smooth manifold, are there any “distinguished” Riemannian metrics?* One interpretation of what it means for a metric to be “distinguished” is to have *constant curvature*. Two important curvatures associated to any Riemannian metric are the *sectional curvature* and the *Ricci curvature*. Riemannian metrics with constant sectional curvature are well-understood: given some standard assumptions, the only Riemannian manifolds with constant sectional curvature are the three model spaces.

An *Einstein metric* is a Riemannian metric with constant Ricci curvature; this is equivalent to the metric satisfying an equation called the *Einstein equation*. Finding Einstein metrics is difficult: one reason for this is that the Einstein equation is a non-linear partial differential equation. Nevertheless, mathematicians have been able to find Einstein metrics is by introducing *symmetry*. Symmetry means picking a group action on our manifold and restricting our search to metrics which are invariant under the group action.

For my research project, I looked at the case when the manifold is R^n or S^n and the group is the *special orthogonal group*, SO(n). I found that this symmetry gives no new examples of Einstein metrics. More precisely, I showed that the only SO(n)-invariant Einstein metrics on R^n or S^n are the ones coming from the three model spaces

**Joseph Kwong**

**The University of Queensland**