I'm asking for big picture in geometry here. I've studied the first three chapters of John Lee's smooth manifolds but still I cannot see the path ahead. My questions are mainly about classification of the subjects of geometry, I like to know beforehand where everything is.I don't need the definitions I just need to see how these fields are connected together and classified . So I list my questions here instead of posting them separately because I think they are related or at least I believe some can make them related:
There are some additional structures on manifolds like Riemannian, Symplectic, complex and so on. How are they classified? Is it correct to say differential geometry is the study of either Riemannian or Symplectic structures or there are finitely other structures? And there are other kinds of structures like complex , quaternionic structures? And where should one place contact geometry, conformal , Möbius geometries?
Where is metric geometry in this picture and what are CAT(k) spaces?
What is the difference between geometric topology and differential topology?
Are G-structures the improved and modernized version of Erlangen program?
Feel free to edit and improve my question and make it CommunityWiki.