All Questions
Tagged with sub-riemannian-geometry lie-groups
12 questions
10
votes
1
answer
706
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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
3
votes
0
answers
143
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Homogeneous Carnot group, its Lie algebra and Carnot-Carathéodory ball
Background: Let the smooth vector fields $X=(X_1,\cdots,X_m)$ define on $\mathbb{R}^n$ and they satisfy the following assumption:
(H1): There is a dilation structure
$$\delta_{t}:\mathbb{R}^n\to \...
- 561
9
votes
1
answer
255
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On the diameter of left-invariant sub-Riemannian structures on a compact Lie group
Let $G$ be a compact connected Lie group with Lie algebra $\mathfrak g$ of dimension $m$.
We fix an inner product $\langle\cdot,\cdot\rangle$ on $\mathfrak g$.
We may assume (in case is necessary) ...
- 2,446
1
vote
1
answer
116
views
When are Carnot groups negatively curved and homeomorphic to Euclidean space
When are Carnot groups complete and negatively curved (in the sense of $CAT(\kappa)$ spaces)?
- 5,405
1
vote
1
answer
191
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Doubling constant of Carnot group
This post shows that every Carnot group is a doubling metric space. However, what is its doubling constant?
- 5,405
4
votes
1
answer
413
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Heisenberg groups, Carnot groups and contact forms
The horizontal distribution in the Heisenberg group is the kernel of the standard contact form:
$$
\alpha = dt + 2 \sum_{j=1}^n (x_j \, dy_j - y_j \, dx_j).
$$
Question. Can one describe ...
- 28k
1
vote
0
answers
136
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Asymptotic cone of discrete group of Heisenberg group $\mathbb{H}^3$
Note that $(\mathbb{Z}^2,d_W)$ where $d_W$ is word metric has asymptotic cone $$(\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
15
votes
3
answers
1k
views
Explicit formulas for Carnot-Carathéodory distances on Carnot groups
Let $G$ be a Carnot group (aka stratified group), so that $G$ is a connected and simply connected finite-dimensonal Lie group, whose Lie algebra $\mathfrak{g}$ admits a decomposition $\mathfrak{g} = ...
- 29.8k
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
6
votes
1
answer
3k
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Difference between the Laplacian and the sub-Laplacian of a Lie group
Given a Lie group $G$, what is the difference between the Laplacian $\Delta$ and the sub-Laplacian $\Delta_{sub}$ of $G$. And what are the properties that we lose when going from sub-Laplace to ...
- 650
4
votes
1
answer
234
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dirichlet problem in the heisenberg group
Good morning everybody.
I was looking just for a quick reference to know whether the Dirichlet problem has a solution in the Heisenberg group, that is $\mathbb R^3$ endowed with coordinates $(x,y,z)$ ...
- 277