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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**?2010-11-29
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    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*.2010-11-29
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    @Pete: Sorry, I just quoted from the Wikipedia article, but I assume, it's about "spaces" (http://en.wikipedia.org/wiki/Space_%28mathematics%29)).2010-11-29

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