Questions tagged [sub-riemannian-geometry]
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17 questions with no upvoted or accepted answers
8
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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 ...
- 399
7
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0
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202
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 ...
- 1,914
6
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0
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245
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 ...
- 369
6
votes
0
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184
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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
5
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146
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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 ...
- 193
5
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0
answers
266
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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
4
votes
0
answers
103
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 ...
- 287
4
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0
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156
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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 $$X_1 = \partial_x \ \text{ and } \ X_2 = |...
- 91
3
votes
0
answers
102
views
Geometric interpretation for a connection whose corresponding distribution generates the whole Lie algebras of vector fields
Let we have a connection on a manifold $M$ so it is considered as a distribution on the tangent bundle $TM$ of $M$. The integrability of this distrbution is equivalent to flatness of the connection.
...
- 366
3
votes
0
answers
56
views
Regularity of subelliptic eigenfunction on characteristic domain
Background: Consider the Hörmander vector fields $X=(X_1,\cdots,X_m)$ on $\mathbb{R}^n$, and the associated Dirichlet eigenvalue problem
$$-\Delta u:=\sum_{i=1}^mX_i^*X_iu=\lambda u~~\text{on}~\Omega,~...
- 561
3
votes
0
answers
143
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
$$\delta_{t}:\mathbb{R}^n\to \...
- 561
3
votes
0
answers
222
views
A question about a paper of Bismut and Lebeau
Let $X$ be a Riemannian manifold, and $Y\hookrightarrow X$ be a closed submanifold of $X$ with normal bundle $N$ with the induced metric.
Then near $Y$, we have $$dv_X(y,Z)=k(y,Z)dv_Y(y)dv_{N_y}(Z),$$...
- 1,915
2
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0
answers
74
views
Solution of lift spinor
Let $\pi:M\to X$ be a fibration map between two spin manifolds, i.e. the fiber $\pi^{-1}(x)$ is a manifold, suppose $s:X\to M$ is an embedding. Let $\Phi$ be a solution of Dirac equation, i.e. $D^X\...
- 1,915
1
vote
0
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188
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Good references to understand sub-Riemannian geometry and Heisenberg groups
I'm looking for books and articles to understand a little about the Heisenberg group and sub-Riemannian geometry, specifically why the Heisenberg group is an example of a sub-Riemannian manifold, and ...
- 677
1
vote
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96
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"Quasiconformal" projections from Heisenberg group to the plane
Let $G$ be the 3-dimensional Heisenberg group equipped with its Carnot-Caratheodory subriemannian metric $d_{G}$. Let $U$ be a domain in $G$ of the form $V \times I$, where $V$ is an open subset of $\...
- 179
1
vote
0
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74
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Ahlfors regular path metric defined by a continuous plane field in $\mathbb{R}^{3}$
Suppose I have a uniformly Holder continuous plane field $H$ on $\mathbb{R}^{3}$. I will assume that this plane field $H$ has many special properties, all of which are completely unreasonable to ...
- 179
1
vote
0
answers
136
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Asymptotic cone of discrete group of Heisenberg group $\mathbb{H}^3$
Note that $(\mathbb{Z}^2,d_W)$ where $d_W$ is word metric has asymptotic cone $$(\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