What structure must $\mathbb{R}^n$ be equipped with for its structure preserving maps to be exactly the translations? I know that when equipped with a metric, the structure preserving maps will be the isometries, but those include rotations and other transformations.
The translations are clearly a subgroup of the isometries, but I'd like to know what additional structure is needed on $\mathbb{R}^n$ for the structure preserving maps to be exactly the translations.
 A: On top of the given smooth structure, you could impose the additional structure of the frame field $(\frac{\partial}{\partial x_1},\ldots,\frac{\partial}{\partial x_n})$. Every translation preserves this structure, and every map that preserves this structure is a translation.
A: Transformations $f$ preserving the equivalence classes of the relation of lines being parallel are known as dilatations because they consist precisely of the translations and dilations, the non-identity diations being those transformations with exactly one fixed point, and the non-identity translations being those with no fixed points (see below for proof). Since the only isometric dilations are reflections over points, which reverse orientation for odd $n$, we may conclude that for odd $n$ the translations are the isometric orientation-preserving dilatations. Since every dilatation has a line (i.e. $1$-dimensional subspace) to which it restricts (any line through the fixed point for a dilation, any line parallel to the direction of a translation), translations may be characterized as transformations that are isometric orientation-preserving dilatations on any subspace they restrict to.

To see that a dilatation $f$ is either dilations or translation, suppose $R$ is the intersection of two lines through distinct points $P$ and $Q$. Then $f(R)$ would have to be the intersection of the unique lines through $f(P)$ and $f(Q)$ parallel to the lines through $P$ and $R$ respectively. If $S$ is on the line through $P$ and $Q$, then it is not on a line through $P$ and $R$ and is determined by $f(P)$ and $f(R)$.
In particular, if $f$ has at least two distinct fixed points $P$ and $Q$, then it is the identity transformation (the zero translation). Moreover, if $f$ has two distinct non-fixed points $P$ and $Q$, then the lines through $P$ and $f(P)$, and through $Q$ and $f(Q)$ intersect in a point $R$ if and only if $R$ is a (necessarily unique) fixed point. Otherwise, those lines do not intersect but join parallel lines through $P$ and $Q$, and $f(P)$ and $f(Q)$, so are non-intersecting lines in the same plane, so are parallel.
Consequently, translations have the property that if $P$ is distinct from $Q$, then $f(Q)$ is the intersection of the line through $f(P)$ parallel to the line throuhg $P$ and $Q$, and the line through $Q$ parallel to the line through $P$ and $f(P)$. In other words, translations have the property that lines through $P$ and $f(P)$ for each point $P$ are parallel, i.e. they are induced by a function within some equivalence class of parallel lines.
