Questions tagged [sub-riemannian-geometry]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
1 vote
0 answers
132 views

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 ...
Ilovemath's user avatar
  • 575
8 votes
1 answer
668 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-...
Piotr Hajlasz's user avatar
3 votes
0 answers
102 views

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. ...
Ali Taghavi's user avatar
5 votes
0 answers
110 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 ...
Theo Diamantakis's user avatar
1 vote
0 answers
79 views

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 \begin{equation}\label{critical} -(\...
sorrymaker's user avatar
4 votes
0 answers
79 views

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 ...
pxchg1200's user avatar
  • 265
3 votes
0 answers
49 views

Regularity of subelliptic eigenfunction on characteristic domain

Background: Consider the Hörmander vector fields $X=(X_1,\cdots,X_m)$ on $\mathbb{R}^n$, and the associated Dirichlet eigenvalue problem $$-\Delta u:=\sum_{i=1}^mX_i^*X_iu=\lambda u~~\text{on}~\Omega,~...
Houa's user avatar
  • 561
7 votes
0 answers
177 views

Why do people who work in stochastic analysis and geometry tend to work in sub Riemannian geometry?

There is a rich theory of diffusions on manifolds. Every time I see someone who studies diffusions on manifolds, it seems like they study the sub Riemannian setting. I get that this is more general ...
user479223's user avatar
  • 1,247
5 votes
1 answer
170 views

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. $g$ ...
Athena's user avatar
  • 275
6 votes
1 answer
484 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 $C_d\geq 0$ ...
Carlos_Petterson's user avatar
7 votes
1 answer
216 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 ...
Son Gohan's user avatar
  • 215
3 votes
0 answers
129 views

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 \...
Houa's user avatar
  • 561
7 votes
2 answers
294 views

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 ...
Jean DELI's user avatar
  • 137
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 ...
Mathsfreak's user avatar
2 votes
1 answer
158 views

A metric naturally arise from the Euclidean symplectic structure?

For $n>1$ let $\omega=\sum_{i=1}^n dx_i\wedge dy_i$ be the standard symplectic structure on $\mathbb{R}^{2n}=\mathbb{R}^n \times \mathbb{R}^n$. We define the following distribution $D$ on $\mathbb{...
Ali Taghavi's user avatar
0 votes
1 answer
138 views

Signature Map From $p$-Geometric Rough Paths to $T(\mathbb{R})$

Let $f:[0,T]\rightarrow \mathbb{R}^d$ be a p-geometric rough path and let $\mathcal{G}_p^d$ be the collection of all such paths. Does the Lyons signature map define a continuous bijection between $\...
ABIM's user avatar
  • 5,069
8 votes
1 answer
416 views

Is $C^{\infty}(M)$ dense in weighted Sobolev space $W_{X}^{1}(M)$?

Let $M$ be a compact manifold without boudary and let $X_{1},\ldots,X_{m}$ be smooth vector fields on $M$. Consider the following weighted Sobolev space: $$ W_{X}^{1}(M)=\{f\in L^{2}(M)|X_{j}f\in L^2(...
pxchg1200's user avatar
  • 265
9 votes
1 answer
244 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) ...
emiliocba's user avatar
  • 2,279
1 vote
1 answer
103 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)?
ABIM's user avatar
  • 5,069
1 vote
1 answer
177 views

Doubling constant of Carnot group

This post shows that every Carnot group is a doubling metric space. However, what is its doubling constant?
ABIM's user avatar
  • 5,069
20 votes
1 answer
1k 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 "...
Piotr Hajlasz's user avatar
4 votes
1 answer
360 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 ...
Piotr Hajlasz's user avatar
2 votes
0 answers
73 views

Solution of lift spinor

Let $\pi:M\to X$ be a fibration map between two spin manifolds, i.e. the fiber $\pi^{-1}(x)$ is a manifold, suppose $s:X\to M$ is an embedding. Let $\Phi$ be a solution of Dirac equation, i.e. $D^X\...
DLIN's user avatar
  • 1,905
3 votes
0 answers
218 views

A question about a paper of Bismut and Lebeau

Let $X$ be a Riemannian manifold, and $Y\hookrightarrow X$ be a closed submanifold of $X$ with normal bundle $N$ with the induced metric. Then near $Y$, we have $$dv_X(y,Z)=k(y,Z)dv_Y(y)dv_{N_y}(Z),$$...
DLIN's user avatar
  • 1,905
1 vote
0 answers
95 views

"Quasiconformal" projections from Heisenberg group to the plane

Let $G$ be the 3-dimensional Heisenberg group equipped with its Carnot-Caratheodory subriemannian metric $d_{G}$. Let $U$ be a domain in $G$ of the form $V \times I$, where $V$ is an open subset of $\...
Clark's user avatar
  • 179
1 vote
1 answer
175 views

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 ...
Ali Taghavi's user avatar
1 vote
0 answers
72 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 ...
Clark's user avatar
  • 179
1 vote
0 answers
122 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 ...
Hee Kwon Lee's user avatar
  • 1,060
14 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} = ...
Nate Eldredge's user avatar
2 votes
1 answer
143 views

Integrability of direct sum of some integrable distributions

Let $M$ be a smooth manifold and let $\Delta _i$ for $i=1,...,k$ be distributions of $TM$ which are integrable such that $\Delta_i \cap \Delta _j$ is zero distribution for $i \neq j$. Suppose $\Delta ...
user103173's user avatar
3 votes
1 answer
566 views

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 ...
pxchg1200's user avatar
  • 265
2 votes
1 answer
284 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 ...
pxchg1200's user avatar
  • 265
4 votes
0 answers
153 views

Estimation on Carnot-Carathéodory metric induced on $\mathbb{R}^3$ by Martinet vector fields

At page 763 of Trace theorems for vector fields - R. Monti, D. Morbidelli the Carnot-Carathéodory metric $d$ induced on $\mathbb{R}^2$ by the vector fields $$X_1 = \partial_x \ \text{ and } \ X_2 = |...
gangrene's user avatar
0 votes
1 answer
82 views

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.)...
Loreno Heer's user avatar
8 votes
1 answer
327 views

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 ...
symplectik3's user avatar
6 votes
2 answers
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 ...
Mizar's user avatar
  • 3,086
2 votes
1 answer
132 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 ...
quyhktn-qa's user avatar
4 votes
2 answers
219 views

Is $\mathbb{P}T^*M$ a sub-Riemannian manifold if $M$ is Riemannian?

(this question is about a particular aspect of a previous question, which was not duly stressed) Let $(M,g)$ a Riemannian $n$-dimensional manifold, and let $$ \widetilde{M}:=\mathbb{P}T^*M $$ be the $...
Giovanni Moreno's user avatar
6 votes
1 answer
2k views

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 ...
Z. Alfata's user avatar
  • 640
5 votes
0 answers
255 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 ...
Raziel's user avatar
  • 3,143
4 votes
1 answer
462 views

Is this distribution completely non integrable?

We consider the usual Riemannian metric on $S^{n}$. Its corresponding LC connection gives us a distribution on $TS^{n}$. Is this distribution completely nonintegrable? In general, what type of ...
Ali Taghavi's user avatar
4 votes
1 answer
217 views

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 $(x,y,z)$ ...
guido giuliani's user avatar
6 votes
0 answers
225 views

On Holonomy in (regular) Riemannian Foliations

Right now, I am trying to understanding the role of holonomy fields on Riemannian foliations, which lead me to the following (probably topological) groupoid: Let $\mathcal{F}\subset M$ be a ...
Llohann's user avatar
  • 369
2 votes
2 answers
389 views

Principal bundles and Subriemannian Geometry

In sub-Riemannian geometry, one considers manifolds $P$ equipped with a subbundle $\mathcal{H}$ of $TP$, the horizontal distribution. One then has a Riemannian metric only on this distribution $\...
Matthias Ludewig's user avatar
4 votes
1 answer
347 views

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 ...
Nikita Evseev's user avatar
8 votes
0 answers
412 views

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 ...
Nikita Evseev's user avatar
15 votes
1 answer
394 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 ...
Anton Petrunin's user avatar
3 votes
2 answers
290 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 ...
Nikita Evseev's user avatar
9 votes
2 answers
684 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 ...
Sandeep Thilakan's user avatar
6 votes
0 answers
181 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 ...
Jon Middleton's user avatar