All Questions
Tagged with sub-riemannian-geometry mg.metric-geometry
18 questions
10
votes
1
answer
706
views
Where to find English translation of Pansu's paper from Ann. Math?
Where can I find English translation of the following paper?
P. Pansu,
Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un. (French. English summary) [Carnot-...
- 28k
5
votes
0
answers
146
views
What do the Carnot groups act on?
My question is in some sense a less ambitious version of the following MO question where the answer was inconclusive.
A Carnot group of step $N$ can be identified within the tensor algebra, modulo ...
- 193
6
votes
1
answer
551
views
Relationship between doubling constant of a metric space and of a metric measure space
Let $(X,d,m)$ be a metric measure space. We say that it is doubling in the sense of metric spaces if for every:
$x\in X$ and every $r>0$ there exists some (metric) doubling constant $C_d\geq 0$ ...
- 195
7
votes
1
answer
246
views
Currents in sub-Riemannian geometry
Federer and Fleming's notion of "currents" is well established so far, and starting from the seminal work of Ambrosio and Kirchheim, the notion of metric currents is well studied also. The ...
- 215
6
votes
3
answers
1k
views
Proof of Rashevskii-Chow theorem
I'm looking for a good quotation and comprehensive explaination of the theorem of Chow-Rashewski.
I'm writing my thesis on sub-Riemannian Geometry and a special control problem. Therefore I want to ...
- 73
9
votes
1
answer
255
views
On the diameter of left-invariant sub-Riemannian structures on a compact Lie group
Let $G$ be a compact connected Lie group with Lie algebra $\mathfrak g$ of dimension $m$.
We fix an inner product $\langle\cdot,\cdot\rangle$ on $\mathfrak g$.
We may assume (in case is necessary) ...
- 2,446
1
vote
1
answer
116
views
When are Carnot groups negatively curved and homeomorphic to Euclidean space
When are Carnot groups complete and negatively curved (in the sense of $CAT(\kappa)$ spaces)?
- 5,405
1
vote
1
answer
191
views
Doubling constant of Carnot group
This post shows that every Carnot group is a doubling metric space. However, what is its doubling constant?
- 5,405
1
vote
0
answers
74
views
Ahlfors regular path metric defined by a continuous plane field in $\mathbb{R}^{3}$
Suppose I have a uniformly Holder continuous plane field $H$ on $\mathbb{R}^{3}$. I will assume that this plane field $H$ has many special properties, all of which are completely unreasonable to ...
- 179
1
vote
0
answers
136
views
Asymptotic cone of discrete group of Heisenberg group $\mathbb{H}^3$
Note that $(\mathbb{Z}^2,d_W)$ where $d_W$ is word metric has asymptotic cone $$(\mathbb{R}^2,\| \ \|_1)=\lim_{t>0\rightarrow 0}\ t(\mathbb{Z}^2,d_W)$$
And Heisenberg group $\mathbb{H}^3$ has an ...
- 1,100
6
votes
2
answers
314
views
Are rays in Carnot groups straight?
A famous open problem in Geometric Control Theory and in the study of sub-Riemannian manifolds is whether constant-speed length minimizers in a sub-Riemannian manifold are always smooth (see also this ...
- 3,146
2
votes
1
answer
135
views
Volume of the subelliptic ball
Let $\Omega \in \mathbb{R}^n$ a bounded open set when $n\geq 2$, and let $X_{1},X_{2},\cdots,X_{m}$ be real smooth vector fields that satisfy Hormander condition on $\Omega$. If we denote $Q(x)$ as ...
- 21
5
votes
0
answers
266
views
Regularity of the distance from the boundary in singular riemannian manifolds
I am looking for references related with the regularity of the distance from the boundary in singular Riemannian manifolds.
I assume the following setting. $(M,g)$ is a Riemannian manifold, with ...
- 3,223
15
votes
1
answer
413
views
bi-Lipschitz gluing
Let $H$ be the Heisenberg group with
left invariant sub-Riemannian metric and $\varepsilon>0$ is small.
Let us denote by $|x-y|_H$ the distance from $x$ to $y$ in $H$.
I have a bi-Lipschitz ...
- 45k
3
votes
2
answers
295
views
Length of non-horizontal curve
Let $M$ be a sub-Riemannian space.
Consider a smooth curve $\gamma:[0,1]\to M$ such that
$\dot\gamma(t)\not\in H_{\gamma(t)}$, where $H_{\gamma(t)}$ is the horizontal subbundle ( i.e. $\gamma$ is ...
- 399
9
votes
2
answers
714
views
Why is proving $C^{\infty}$ regularity of sub Riemannian geodesics so hard?
In Montgomery's A Tour of Subriemannian Geometries, Their Geodesics and Applications, problem 10.1 in Chapter 10 asks "Is every minimizing geodesic smooth ?".
Can someone explain what are the major ...
- 317
6
votes
0
answers
184
views
The Tangent Bundle of the Space of CR Structures on S^(2n+1)
Let $M$ be a smooth compact $n$-manifold without boundary, $g$ some choice of Riemannian metric on $M$, and $\omega_g$ the volume form gotten from $g$. Say you're interested in finding extrema for ...
- 1,101
4
votes
3
answers
1k
views
Ball-Box Theorem and Sequence of Distributions
Let $(e^k,g^k)$ be a sequence of 2d smooth distributions in $R^3$ (with Euclidean metric) s.t $e^k,g^k$ are orthogonal. Let $f^k$ normal direction to this distribution. Suppose $[e^k,g^k] \neq 0 $ on ...
- 419