Geodesic metrics that admit dilatation at each point

Question

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 $\mathbb CP^n$.

Answer 1

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 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.

Answer 2

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.