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I'm looking for a book where the Erlangen program is carried out on some example groups with explicit computations.

What I mean by "carrying out Erlangen program" is picking a specific group (say SO(n+1)) and some subgroup (say SO(n)), finding all invariants and their relations (and the relations between these relations and so on), and see what it means for the geometry of the natural space our group acts on (say n-sphere). So all the work should be done in the group (including finding good parametrizations for geodesics and this kind of things) rather than in the usual space.

Anyone knows of a good reference?

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This is a very expansive interpretation of the Erlangen program: ordinarily, one picks a space $X$ and a symmetry group $G$ (usually, highly transitive), and describes geometry of $X$ via multipoint invariants, i.e. functions $X^n\to\mathbb{R}$ invariant under the diagonal action (with implicit choice of the class of functions). So, if understand the question correctly, your space is $X=SO(n+1)/SO(n)$ (which is the $n$-sphere) and the symmetry group is $G=SO(n+1).$ However, one needs to specify the class of functions, and geodesics refer to Riemannian connection, which is not group-theoretic. – Victor Protsak Jul 19 '11 at 18:55

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