Curvature and Holes

Quinn Patterson, University of Wollongong

My project has been concerned with the relationship between curvature and holes. The first question we must ask then is what do we mean by curvature? For a surface, we can use a nice picture:

A surface has positive curvature at a point if all the curves passing through that point bend in the same direction. The curvature is negative at that point if there are curves that pass through the point bending in opposite directions like a saddle. The curvature is positive if there is some curve that passes through the point which does not bend at all.

For example the sphere has positive curvature everywhere. The Torus (a doughnut) can be seen to have all three curvatures at different points. On the outside curves bend in the same direction, whilst inside the hole the Torus looks like one big saddle with curves bending in opposite directions.

Now we wish to relate the curvature of a surface to the existence of holes. But how do we come up with a good notion of a hole? To keep track of holes, we count how many kinds of loops there are that can’t be pulled into a point without having to be broken or having to leave the surface. Imagine you laid down a loop of string on the sphere. If you pulled both ends of the string tighter so that the loop contracted smaller and smaller, it would eventually just become a point, and all this can be done without having to leave the sphere or having to break the string. On the Torus however, there are two different loops that cannot be pulled off to a point and cannot be slid into one another. These are the two different types of black lines in the picture above. If you tied a piece of string around through the hole then you could slide it around the cylinder of the Torus, but there would be no way to pull the string into a point because the cylinder is in the way! The other kind of loop would be if you tied a string along the top of the Torus around the hole. You wouldn’t then  be able to contract the string to a point without having to leave the surface entirely! You also cannot slide the two different types of loops into each other either. If your surface had any kind of hole, you would be able to tie a loop through the hole which then could not be slipped off the surface. Thus if we have loops that can’t be contracted, we must have a hole.

The main result of my project has been using this notion of a ‘hole’ to prove the following:

Theorem: If a surface has a hole, then it must have negative curvature at some point.

There are quite a few technicalities and caveats. Mathematics allows us to consider surfaces in ‘higher dimensions’, and in higher dimensions there are many more directions you can bend, making curvature a lot more complicated. Similarly, loops which can’t be contracted to points behave strangely. However, with the intuition of the case for surfaces, in my project we have been able to show similar analogous statements for surfaces in higher dimensions too.

Quinn Patterson was one of the recipients of a 2017/18 AMSI Vacation Research Scholarship.

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