# Questions tagged [sub-riemannian-geometry]

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### Conformal equivalence degenerate metric tensors

Assume that $g$ and $g'$ are metric tensors with one dimensional kernel and the same signature. Does the classical results of Weyl (dim >3) or of Cotton (dim=3) generalise to that case, i.e. $g$ ...
341 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 $C_d\geq 0$ ...
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### 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 ...
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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) ...
1 vote
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### 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)?
1 vote
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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?
794 views

### Carnot-Carathéodory metric

The metric in sub-Riemannian geometry is often called the Carnot-Carathéodory metric. Question 1. What is the origin of this name? Who was the first to introduce it? I believe that the "...
255 views

### 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 ...
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1 vote
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### A possible sub-Riemannian structure associated to a non-symmetric matrix

Let $A=(a_{ij})$ be an invertible matrix with real entries $a_{ij}$. We associate to $A$ the $1$-form $\alpha=\sum_i (\sum_j a_{ij}x_j)dx_i$. The distribution $\ker \alpha$ is integrable if and ...
1 vote
64 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 ...
1 vote
98 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 ...
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### how to use the sobolev inequality to obtain the embedding theorem

I am reading Luca Capogna's article An Embedding theorem and the Harnack inequalitiy for nonlinear subelliptic equations. In this article, the authors proved the following theorem (Theorem 2.3) Let ...
264 views

### The relationship about sub-unit ball and sub-elliptic ball

Let $\Omega$ be a bounded open domain in $\mathbb{R}^{n}$ with smooth boundary.$\{X_{1},\cdots,X_{m}\}$ be smooth real vector fields on $\Omega$ Which satisfy the Hormander condition. If $\gamma$ is ...
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343 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 ...
281 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 ...
647 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 ...
166 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 ...
328 views

### Converse to Chow's theorem in sub-riemannian geometry

Chow's theorem is the statement that if $M$ is a connected smooth manifold endowed with a distribution $\mathcal{D}$ which is completely non integrable (i.e. iterated commutators of smooth sections of ...
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 ...