# Euclidean geometry and the Euclidean group

At Wikipedia's Erlangen program I read that "quite often, it appears there are two or more distinct geometries with isomorphic automorphism groups". Some examples are given.

But what are examples of geometries nevertheless being uniquely defined (up to isomorphism) by the isomorphism class of their automorphism group?

Are there examples of non-Euclidean geometries with an automorphism group isomorphic to the Euclidean group?

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What is your definition of a geometry? –  Pete L. Clark Nov 29 '10 at 3:01
It sounds like you're looking for a Lie group $G$ that acts on spaces $X_1$ and $X_2$ such that the point stabilizers $G_p$ act faithfully on $T_pX_i$ for $p \in X_i$, and you want any two such actions to always be conjugate. Since $G$ always acts on itself, this means it can't have any proper closed subgroups, so it has to be $S^1$, so I think the answer to your question is no. –  Ryan Budney Nov 29 '10 at 6:10
@Pete: Sorry, I just quoted from the Wikipedia article, but I assume, it's about "spaces" (en.wikipedia.org/wiki/Space_%28mathematics%29)). –  Hans Stricker Nov 29 '10 at 7:12