I have often heard that Lie groups classify geometry. For example that $O(n)$ is about real manifolds, $U$ is about almost complex manifolds, $SO(n)$ about orientable real manifolds and so on.


I have also heard that manifolds can be defined very generally by patching local pieces via a pseduogroup of morphisms. 


My questions are 


1) Does this pseduogroup relate to the Lie group?


2) How do Lie groups classify geometry?


3) Is there a geometry for every Lie group or only some of them? 


and maybe even


4) Does this classification seem work for algebraic (i.e. non-differential) geometry?


I am wondering whether this Erlangen program classification is actually formalized, or only serves as a slogan or principle.



How do Lie groups classify geometry?
[R.W.Sharpe, Differential Geometry - Cartan's Generalization of Klein's Erlangen Program](http://books.google.com/books?id=Ytqs4xU5QKAC&pg=PA173&dq=cartan%27s+generalization+of+klein%27s+erlangen&hl=it&ei=aGK4TPfXLIPKswbmsfyqDQ&sa=X&oi=book_result&ct=result&resnum=1&ved=0CCYQ6AEwAA#v=onepage&q&f=false)


