# Questions tagged [sub-riemannian-geometry]

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### Good references to understand sub-Riemannian geometry and Heisenberg groups

I'm looking for books and articles to understand a little about the Heisenberg group and sub-Riemannian geometry, specifically why the Heisenberg group is an example of a sub-Riemannian manifold, and ...
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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-...
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### Geometric interpretation for a connection whose corresponding distribution generates the whole Lie algebras of vector fields

Let we have a connection on a manifold $M$ so it is considered as a distribution on the tangent bundle $TM$ of $M$. The integrability of this distrbution is equivalent to flatness of the connection. ...
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### 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 ...
1 vote
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### positivity of solution for Baouendi-Grushin operator

Let $(x,y)\in\mathbb{R}^N = \mathbb{R}^m \times \mathbb{R}^n$ with $m \geq 1$, $n\geq 0$, $\alpha \geq 0$ and $Q=m+ n(\alpha +1)$. Consider the following eqution \label{critical} -(\...
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### Characteristic of Sobolev space generated by Hörmander vector fields

Let $\Omega$ be an open bounded domain in $\mathbb{R}^{n}$ with smooth boundary $\partial\Omega$. Suppose that $X=(X_{1},X_{2},\ldots,X_{m})$ are smooth vector fields defined on $\mathbb{R}^{n}$ and ...
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### How to find equations of a sub-Riemannian problem

I am working on sub-Riemannian geometry and try to understand what are the tools to find the equations of a sub-Riemannian problem. Here is an example: Let us consider the system defined by a ...
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### 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 ...
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### quasi-conformal embedding of Carnot group into euclidean space

By Pansu's theorem, there are no bi-Lipschitz embeddings of Carnot groups (with exception of the Euclidean space itself) into Euclidean space. Do there exist quasi-conformal embeddings (into Eucl. sp.)...
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### Horizontal distribution of a totally geodesic foliation

Let $\mathbb{M}$ be a $n+m$ dimensional manifold. Consider on $\mathbb{M}$ a rank $n$ sub-bundle $\mathcal{H}$ of the tangent bundle. We assume that $\mathcal{H}$ is endowed with a fiber wise inner ...
306 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 ...
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### 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 ...
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### Horizontal Sobolev space on Carnot group

This question is connected with my previous: Heisenberg group: function without vertical derivative. Here I am trying to look from another side: what is a difference between Sobolev space and ...
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### Heisenberg group: function without vertical derivative

Let $\mathbb H$ be Heisenberg group with vector fields $$X=\partial_x - \frac12y\partial_t,\quad Y=\partial_y + \frac12x\partial_t,\quad T=\partial_t$$ and $U\subset\mathbb H$ is an open set. I am ...
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### 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 ...
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### 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 ...
### Why is proving $C^{\infty}$ regularity of sub Riemannian geodesics so hard?
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 ...