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I posted this question at math.stackexchange.com but didn't get an answer. Is it a dumb question, eventually?

There are three ways of characterizing the abstract Euclidean space $E^n$ that are quite different in spirit:

  1. axiomatically (with axioms concerning dimension)
  2. by the abstract Euclidean group $E(n)$ (as its symmetry group, determining $E^n$ uniquely)
  3. by presupposing a metric and requiring that the space is a maximal one with respect to the property that the $(n+1)$-dimensional Cayley-Menger determinant vanishes for all $k$-tuples of points for $k \geq n+2$ and does not vanish for all $k$-tuples of points "in general position" for $k < n+2$.

Question 1: Is it correct, actually, that $E^n$ is uniquely determined by 2 and 3?

Question 2: What are still other ways of characterizing $E^n$ "different in spirit"?

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It is probably best if you specify a category in which you are working, e.g., topological spaces, or smooth manifolds. –  S. Carnahan Dec 1 '10 at 8:55
    
@Scott: You mean, if I specified such a category, you'll give me another characterization? –  Hans Stricker Dec 1 '10 at 9:30
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Its the only complete, connected, simply connected, Riemannian manifold of non-positive curvature. en.wikipedia.org/wiki/Cartan%E2%80%93Hadamard_theorem –  Otis Chodosh Dec 1 '10 at 9:54
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@Otis: This is only up to diffeomorphism. Metrically, hyperbolic $n$-space and Euclidean space are completely different. –  Theo Buehler Dec 1 '10 at 10:19

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