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 $\mathbb CP^n$.



Geodesic metrics that admit dilatation at each point
Concerning the first question: you description is incomplete, even in the homogeneous case.


There are homogeneous geodesic metrics that admit smooth families of dilatations but are not made of flat Banach metrics. In particular, some [Carnot-Caratheodory metrics](http://en.wikipedia.org/wiki/Sub-Riemannian_manifold) are.


For example, consider the Heisenberg group $H$, which can be thought of as $\mathbb R^3$ equipped with the following group law:
$$
 (x,y,z)\cdot(x',y',z') = (x+x',y+y',z+z'+x'y) .
$$
Observe that for every $t\in\mathbb R$, the map $\phi\_t:(x,y,z)\mapsto (e^tx,e^ty,e^{2t}z)$ is a group homomorphism, and these maps form a smooth 1-parameter group of diffeomorphisms (and hence a flow generated by a smooth vector field).


Consider a left-invariant two-dimensional distribution $V\subset TH$ spanned by left-invariant vector fields $X$ and $Y$ whose values at $(0,0,0)$ equal $\partial/\partial x$ and $\partial/\partial y$, respectively. Equip this distribution with a left-invariant Euclidean metric. The distribution is completely non-integrable, so we get a Carnot-Caratheodory metric on $H$. Observe that $\phi\_t$ maps $X$ to $e^tX$ and $Y$ to $e^tY$, hence it is a $e^t$-dilatation of the Carnot-Caratheodory metric.


The Carnot-Caratheodory metric is very different from Banach metrics. For example, its Hausdorff dimension equals 4.


