All Questions
9 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
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
$$\delta_{t}:\mathbb{R}^n\to \...
- 561
2
votes
1
answer
158
views
A metric naturally arise from the Euclidean symplectic structure?
For $n>1$ let $\omega=\sum_{i=1}^n dx_i\wedge dy_i$ be the standard symplectic structure on $\mathbb{R}^{2n}=\mathbb{R}^n \times \mathbb{R}^n$.
We define the following distribution $D$ on $\mathbb{...
- 366
3
votes
0
answers
222
views
A question about a paper of Bismut and Lebeau
Let $X$ be a Riemannian manifold, and $Y\hookrightarrow X$ be a closed submanifold of $X$ with normal bundle $N$ with the induced metric.
Then near $Y$, we have $$dv_X(y,Z)=k(y,Z)dv_Y(y)dv_{N_y}(Z),$$...
- 1,915
2
votes
1
answer
148
views
Integrability of direct sum of some integrable distributions
Let $M$ be a smooth manifold and let $\Delta _i$ for $i=1,...,k$ be distributions of $TM$ which are integrable such that $\Delta_i \cap \Delta _j$ is zero distribution for $i \neq j$. Suppose $\Delta ...
- 23
2
votes
1
answer
292
views
The relationship about sub-unit ball and sub-elliptic ball
Let $\Omega$ be a bounded open domain in $\mathbb{R}^{n}$ with smooth boundary.$\{X_{1},\cdots,X_{m}\}$ be smooth real vector fields on $\Omega$ Which satisfy the Hormander condition. If $\gamma$ is ...
- 287
4
votes
2
answers
220
views
Is $\mathbb{P}T^*M$ a sub-Riemannian manifold if $M$ is Riemannian?
(this question is about a particular aspect of a previous question, which was not duly stressed)
Let $(M,g)$ a Riemannian $n$-dimensional manifold, and let
$$
\widetilde{M}:=\mathbb{P}T^*M
$$
be the $...
- 2,527
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