SO(n)-invariant Einstein metrics

Smooth manifolds are generalisations of smooth curves and smooth surfaces. A (Riemannian) metric is a mathematical object defined on a smooth manifold which gives us notions of angles, distances, and curvatures.  Geometers have been interested in the following question: is there a “best” metric on a given manifold? 

One interpretation of “best” metrics are metrics having constant curvature. There are three main types of curvature associated with any metric: sectional, Ricci, and scalar. Constant sectional curvature metrics are classified: the only ones are the Euclidean metric and the hyperbolic metric on R^n, and the round metric on S^n (assuming simply-connectedness and completeness). Thus, it is natural to next study metrics with constant Ricci curvature. A metric with constant Ricci curvature is called Einstein.

The aim of this project is to write down proofs of the following well-known facts: (1) the only (complete) Einstein metrics on R^n which are invariant under rotations are the Euclidean metric and the hyperbolic metric; (2) The only Einstein metric on S^n which is invariant under rotations fixing an axis is the round metric.

Joseph Kwong

The University of Queensland

Joseph Kwong is a third-year mathematics student at the University of Queensland. Joseph is interested in pure mathematics. In particular, he is interested in differential geometry. Joseph plans to do an honours project in differential geometry in 2024.

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