A Riemannian manifold is called Einstein if its Ricci curvature is a constant multiple of the metric. An example is the n-sphere equipped with the round metric. An interesting question is whether or not we can put other Einstein metrics on n-spheres. In a recent paper, Neinhaus and Wink constructed non-round Einstein metrics on the ten sphere. A crucial step in the construction is by considering doubly warped Einstein metrics on a particular dense open subset M, then extending the metric to the entire sphere. It is well known that under certain boundary conditions, doubly warped metrics on M can be extended smoothly. Neinhaus and Wink claim in their paper that the Einstein condition significantly simplifies these boundary conditions, but a proof is not given. The main aim of our project is to write a proof of this claim. On the way, we wish to understand the boundary conditions for which doubly warped metrics on M extend smoothly to the entire sphere, and how the Einstein condition changes these boundary conditions.
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.