I posted this question at [math.stackexchange.com](https://math.stackexchange.com/questions/12487/characterizations-of-euclidean-space) 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"?
> 
> 
> 



Characterizations of Euclidean space
Herbert Busemann provided many metric characterizations of the elementary spaces in his 1955 book *The Geometry of Geodesics*. To characterize $E^n$, we can then further restrict to cases with zero curvature, or to non-compact cases with non-negative curvature, or probably in several other ways.


For a first pass at these characterizations, we can think of them as being about complete connected Riemannian manifolds; the results actually hold for the more general [G-spaces](https://mathoverflow.net/questions/34394/g-spaces-and-manifolds) instead. In any case, here are five of the characterizations in that book.


**Theorem 15.4, via Desargues's theorem** (p. 87): In a Riemannian G-space, if the geodesic through two points is unique, and any three points lie in a plane, then the space is either Euclidean, hyperbolic, or spherical.


**Theorem 47.4, via bisectors** (p 331): If each bisector $B(a,a')$ (i.e. the locus $xa=xa'$) of a G-space contains with any two points $x,y$ at least one geodesic segment between them, then the space is Euclidean, hyperbolic, or spherical of dimension greater than 1.


**Theorem 48.8, via motions of three points** (p. 337): If a G-space possesses for any four points $a,a',b,c$ with $ab=a'b$ and $ac=a'c$ a motion which leaves $b$ and $c$ fixed and carries $a$ in to $a'$, then the space is Euclidean, hyperbolic, or spherical.


**Theorem 49.7, via reflections** (p. 347): A G-space which can be reflected in each lineal element is Euclidean, hyperbolic, spherical or elliptic. 


**Theorem 55.3, via doubly transitive motions** (p. 395): A G-space whose dimension is finite and odd (or two) and which possesses a pairwise transitive group of motions is Euclidean, hyperbolic, spherical or elliptic.


There are still more characterizations in section 24, via the parallel postulate, and in chapter VI.


