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 $\mathbb CP^n$.