All Questions
Tagged with sub-riemannian-geometry dg.differential-geometry
27 questions
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
$$-\Delta u:=\sum_{i=1}^mX_i^*X_iu=\lambda u~~\text{on}~\Omega,~...
- 561
5
votes
1
answer
185
views
Conformal equivalence degenerate metric tensors
Assume that $g$ and $g'$ are metric tensors with one dimensional kernel and the same signature.
Does the classical results of Weyl (dim >3) or of Cotton (dim=3) generalise to that case, i.e. $g$ ...
- 275
7
votes
1
answer
246
views
Currents in sub-Riemannian geometry
Federer and Fleming's notion of "currents" is well established so far, and starting from the seminal work of Ambrosio and Kirchheim, the notion of metric currents is well studied also. The ...
- 215
3
votes
0
answers
143
views
Homogeneous Carnot group, its Lie algebra and Carnot-Carathéodory ball
Background: Let the smooth vector fields $X=(X_1,\cdots,X_m)$ define on $\mathbb{R}^n$ and they satisfy the following assumption:
(H1): There is a dilation structure
$$\delta_{t}:\mathbb{R}^n\to \...
- 561
7
votes
2
answers
312
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 ...
- 139
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{...
- 366
8
votes
1
answer
498
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:
$$ W_{X}^{1}(M)=\{f\in L^{2}(M)|X_{j}f\in L^2(...
- 287
9
votes
1
answer
255
views
On the diameter of left-invariant sub-Riemannian structures on a compact Lie group
Let $G$ be a compact connected Lie group with Lie algebra $\mathfrak g$ of dimension $m$.
We fix an inner product $\langle\cdot,\cdot\rangle$ on $\mathfrak g$.
We may assume (in case is necessary) ...
- 2,446
2
votes
0
answers
74
views
Solution of lift spinor
Let $\pi:M\to X$ be a fibration map between two spin manifolds, i.e. the fiber $\pi^{-1}(x)$ is a manifold, suppose $s:X\to M$ is an embedding. Let $\Phi$ be a solution of Dirac equation, i.e. $D^X\...
- 1,915
3
votes
0
answers
222
views
A question about a paper of Bismut and Lebeau
Let $X$ be a Riemannian manifold, and $Y\hookrightarrow X$ be a closed submanifold of $X$ with normal bundle $N$ with the induced metric.
Then near $Y$, we have $$dv_X(y,Z)=k(y,Z)dv_Y(y)dv_{N_y}(Z),$$...
- 1,915
1
vote
1
answer
176
views
A possible sub-Riemannian structure associated to a non-symmetric matrix
Let $A=(a_{ij})$ be an invertible matrix with real entries $a_{ij}$.
We associate to $A$ the $1$-form $\alpha=\sum_i (\sum_j a_{ij}x_j)dx_i$.
The distribution $\ker \alpha$ is integrable if and ...
- 366
15
votes
3
answers
1k
views
Explicit formulas for Carnot-Carathéodory distances on Carnot groups
Let $G$ be a Carnot group (aka stratified group), so that $G$ is a connected and simply connected finite-dimensonal Lie group, whose Lie algebra $\mathfrak{g}$ admits a decomposition $\mathfrak{g} = ...
- 29.8k
2
votes
1
answer
148
views
Integrability of direct sum of some integrable distributions
Let $M$ be a smooth manifold and let $\Delta _i$ for $i=1,...,k$ be distributions of $TM$ which are integrable such that $\Delta_i \cap \Delta _j$ is zero distribution for $i \neq j$. Suppose $\Delta ...
- 23
3
votes
1
answer
606
views
how to use the sobolev inequality to obtain the embedding theorem
I am reading Luca Capogna's article An Embedding theorem and the Harnack inequalitiy for nonlinear subelliptic equations. In this article, the authors proved the following theorem
(Theorem 2.3) Let ...
- 287
2
votes
1
answer
292
views
The relationship about sub-unit ball and sub-elliptic ball
Let $\Omega$ be a bounded open domain in $\mathbb{R}^{n}$ with smooth boundary.$\{X_{1},\cdots,X_{m}\}$ be smooth real vector fields on $\Omega$ Which satisfy the Hormander condition. If $\gamma$ is ...
- 287
4
votes
0
answers
156
views
Estimation on Carnot-Carathéodory metric induced on $\mathbb{R}^3$ by Martinet vector fields
At page 763 of Trace theorems for vector fields - R. Monti, D. Morbidelli the Carnot-Carathéodory metric $d$ induced on $\mathbb{R}^2$ by the vector fields $$X_1 = \partial_x \ \text{ and } \ X_2 = |...
- 91
2
votes
1
answer
135
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 ...
- 21
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 $...
- 2,527
6
votes
1
answer
3k
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 ...
- 650
5
votes
0
answers
266
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 ...
- 3,223
4
votes
1
answer
480
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 ...
- 366
2
votes
2
answers
442
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 $\...
- 9,853
3
votes
2
answers
295
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 ...
- 399
9
votes
2
answers
714
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 ...
- 317
4
votes
2
answers
375
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 ...
- 2,479
4
votes
3
answers
1k
views
Ball-Box Theorem and Sequence of Distributions
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 ...
- 419