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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 $\mathbb CP^n$.

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I can't figure out your first sentence. As $v_x$ is said to fix $x,$ this suggests an action or infinitesimal action on $U_x,$ but then $v_x$ acts on the set of metrics. Could you please expand on this a bit, maybe say more about how $\mathbf R^n$ is an example? –  Will Jagy Sep 25 '10 at 18:24
    
@Will, thanks for your remark, that was sloppy indeed, I meant that the flow generated by $v_x$ should dilatate the metric constantly. –  Dmitri Sep 25 '10 at 18:55
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Full marks for using the word "dilatation" but I think "dilatatable" is one syllable too far. –  gowers Sep 26 '10 at 8:17
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Берестовский В. Н. Подобно однородные локально полные пространства с внутренней метрикой. Известия ВУЗов. Математика. — 2004. — № 11(510). — с. 3-22. –  Anton Petrunin Sep 26 '10 at 16:19

2 Answers 2

up vote 9 down vote accepted

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.

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Sergei, thanks a lot for the answer! It makes clear that Banach metrics are not enough, and also that an existence of a polyhderal stratification is too optimistic. Still I wonder if some kind of decent stratified structure always appear for this dilatateble metrics... –  Dmitri Sep 25 '10 at 21:01
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I believe there is one. Let's say that two point are connected if one is an image of the other under one of these local dilatations. Two points are equivalent if they one can get from one to the other via a chain of such connections. It seems that equivalence classes form a stratification into smooth submanifolds, but I did not try to check the details. –  Sergei Ivanov Sep 25 '10 at 21:24
    
Yes, this sounds plausible. I wonder, if you impose the condition, that the metric is complete, $M^n$ is simply connected, and the size of $U_x$ for all $x$ is at least $\varepsilon$ -- can there be classification of all dilatatable metrics in this case? (So this should include flat ones and Carnot-Caratheodory). Do you think this question was studied? Finally I am not sure that I interpret correctly your second phrase -- that "the description is not complete even in homogeneous case". What "homogeneous" means here? The transitivity of isometry group? –  Dmitri Sep 25 '10 at 21:49
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Yes I meant transitivity of the isometry group. If sizes of neighborhoods are bounded away from zero, then all points are equivalent in the sense of my previous comment, hence all these neighborhoods are isometric, so the space is locally homogeneous. I vaguely remember that Berestovskij proved that every homogeneous geodesic metric is a Finsler Carnot-Caratheodory metric. I don't know whether anyone studied which of those admit dilatations but this should not be hard. –  Sergei Ivanov Sep 25 '10 at 22:15
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I think homogeneous examples come from nilpotent Lie groups. The low dimensional ones tend to have expanding automorphisms, but most high-dimensional ones do not. I think the list is complicated. The possible structure near lower-dimensional strata, where size of neighborhoods goes to 0, seems interesting and intricate. But, Maybe it's worth 1st settling it assuming hausdorff dimension is normal. –  Bill Thurston Sep 26 '10 at 7:16

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.

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Marius, thanks a lot! I will have a look. –  Dmitri Nov 9 '10 at 23:15

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