Contents
We will look at hyperbolic spaces.
Systole: A closed curve that is shortest in the hyperbolic metric. (locally or up to isotopy?) Systole function. For each metric inside a teichmüller space the systole function takes on the length of the shortest systole.
It is something of a topological morse function. The function has critical and regular points with similar properties to the morse function. (Akrout, Schmutz)
Systole properties
Systoles can fill the surface. Thurston spine is the set of all points at which the systoles fill. (has a triangulation)
We can define something called a set of minima. This can be used to recognize a critical point. We also define something called a eutactic set of curves.