:I am not sure why you are so keen on adding unsourced discursive essays, which are often just small remarks. It would be extremely helpful if you could prepare a self-contained summary of section 7.2.1 in Berger's panorama on systoles on surfaces. There are are many more results on global differential geometry of surfaces to be found in standard texts which have not been mentioned in the article (e.g. what can be found in Willmore's elementary book on Differential Geometry).