These notes are very preliminary and subject to change considerably without warning!
These are lectures notes for the 2018 AMSI Winter School on curvature. They comprise a brief introduction to comparison geometry in five lectures and are more of a guide with references to the primary source, rather than complete lecture notes. The primary source is:
Comparison Theorems in Riemannian Geometry, J-H. Eschenburg. The figures accompanying the text are here.
Lecture 00 contains background reading in preparation for the course. Students should be more or less familiar with the material discussed there but perhaps not necessarily my idiosyncrasies and approach to differential geometry. Thus it would be a good idea to read Lecture 00 first if only to become accustomed to my style.
More in depth material can be found in my:
These are quite rough and are is sore need of some pictures, but may still be useful - if you do read them, let me know if you spot any typos or errors! Weeks 7 and 8 are very relevant for us and are discussed in Lecture 01 here. Week 10, lecture 02 is the beginning of the comparison theory that we will develop more systematically and in more detail here.