TITLE: Geodesic metrics that admit dilatation at each point
QUESTION [13 upvotes]: Consider the  class of geodesic  metrics $g$ on manifolds, that have the following
property:  for each point $x$ there exists a neighbourhood $U_x$ and 
a smooth vector field $v_x$ in $U_x$ that vanishes at  $x$ and whose flow (for small time) dilatates $g$ by a constant factor.
Let us call such metrics dilatatable.
An obvious example is provided by an Euclidean $\mathbb R^n$, the flow of the field 
$\sum_i x_i \frac{\partial}{\partial x_i}$ dilatates the Euclidean metric by 
a constant factor. More generally one can take any Banach space.
I would like to make a guess about the structure of such metrics in general.
Guess.  Suppose $g$ on $M^n$ is dilatatable. Then there exists a triangulation of $M^n$
such that the restriction of the metric $g$ to each simplex if flat with respect
to the flat structure on the simplex, and 
$g$ is flat on the complement to the union of all  co-dimension $2$ simplexes. 
The first question is the following: was such class of metrics considered somewhere and 
is this guess correct? Are there obvious counterexamples?
Second part of the question is about examples. It is not hard to construct an example 
of such a metric, if we don't require $M^n$ to be a smooth manifold. Namely, we can take any polyhedral metric
on $M^n$, i.e. glue $M^n$ from a union of Euclidean simplexes (glue the boundaries by isometries).
Then for each point there is a conical neighbourhood, and obviously we can always scale this neighbourhood 
by the radial field emanating from $x$. So now comes the
Second question. Take a topological manifold $M^n$ of dimension $n<7$ with such a polyhedral metric. 
It is known then that such a manifold has a smooth structure (because a PL structure in dimension
up to $6$ always defines a unique smooth structure).  Is it possible to chose this smooth structure 
in such a way, that the polyhedral metric is dilatatable for the smooth structure?
The answer to this question is positive for $n=2$, but I don't know already what happen for $n=3$.
At the same time, there are non-trivial examples in higher dimensions, coming from complex geometry.
For example one can quotient some complex tori $\mathbb T^n$ by a finite group of isometries to get 
$\mathbb CP^n$, the obtained polyheral metric on $\mathbb CP^n$ is dilatatable with respect 
to the canonical complex (and hence smooth) structure on $\mathbb CP^n$.

REPLY [2 votes]: Relative to comments by Sergei Ivanov and Bill  Thurston, maybe this line of research concerning "metric spaces with dilations" or "dilation structures" provides a precise 
answer, more general than Berestovskii result.  See this introduction and dig into the biblio. 
Concerning examples related to Carnot-Caratheodory geometry and nilpotent groups (precisely: "Carnot groups"), they appear naturally as models of the (metric) tangent space to a point in a space with dilations. 
If you stand to read a more algebraic account, see emergent algebras, where 
it is proven that this is not really a metric induced phenomenon.