By Timothy Buttsworth, The University of Queensland
Imagine the surface of a sphere. It’s nice and smooth- it has no sharp bits. If we cut it in half along the equator, we get the surfaces of two hemispheres. Now imagine the opposite- if we have the surface of a hemisphere, we can create two copies of it, and attach them along their edges to once again create the surface of a sphere, and of course it is smooth. Finally, imagine if we just start with a tiny slice of the surface of a sphere, say a quarter. If we perform the same ‘doubling’ operation to it, the resulting object is no longer smooth- it has a cusp.
Why did this happen? Both objects were smooth before the doubling, and topologically, they are exactly the same, yet, after the ‘doubling’ procedure, we found one smooth object, and one object with a sharp part. Mathematically, we call all of these objects ‘manifolds’, and we call the half and quarter surfaces of the sphere ‘manifolds with boundary’, because they have an edge or ‘boundary’. In fact, these two objects are actually the same as a ‘manifold with boundary’. The doubling produced different results because we assigned a different ‘Riemannian Metric’ to each.
The Riemannian Metric on a manifold completely determines the shape. The differing Riemannian Metric is why the quarter and half surfaces of a sphere look different, even though they are the same topologically. In 1991, Mori provided two conditions on the Riemannian Metric of a manifold with boundary that were necessary and sufficient to ensure that the ‘doubling’ technique would produce another manifold that looks ‘smooth’.
Knowledge of this technique, particularly Mori’s conditions, is useful for converting problems on a manifold with boundary to a manifold without boundary. As part of my work in the summer of 2014/2015, I used this technique to investigate a process that alters the Riemannian Metric on a manifold. This process is called the Ricci Flow, and this is primarily what Perelman used to prove the famous Poincaré Conjecture in 2003. Like what I did with the Ricci Flow, several powerful results about manifolds without boundary can be extended to manifolds with boundary through this doubling technique.
Timothy Buttsworth was one of the recipients of a 2014/15 AMSI Vacation Research Scholarship.