Characterizations of Euclidean space

Question

^{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"?

Answer 1

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 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.