Heisenberg groups, Carnot groups and contact forms
$begingroup$
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 horizontal distribution in any Carnot group in terms of kernels of some $1$-forms?
I believe the answer should be in the positive and I am looking for references for explicit constructions of such forms so that in the case of the Heisenberg group these constructions would give the contact form.
lie-groups sub-riemannian-geometry heisenberg-groups
asked Feb 10 at 21:51
Piotr HajlaszPiotr Hajlasz
8,86142969
$endgroup$
add a comment |
$begingroup$
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 horizontal distribution in any Carnot group in terms of kernels of some $1$-forms?
I believe the answer should be in the positive and I am looking for references for explicit constructions of such forms so that in the case of the Heisenberg group these constructions would give the contact form.
lie-groups sub-riemannian-geometry heisenberg-groups
asked Feb 10 at 21:51
Piotr HajlaszPiotr Hajlasz
8,86142969
$endgroup$
add a comment |
$begingroup$
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 horizontal distribution in any Carnot group in terms of kernels of some $1$-forms?
I believe the answer should be in the positive and I am looking for references for explicit constructions of such forms so that in the case of the Heisenberg group these constructions would give the contact form.
lie-groups sub-riemannian-geometry heisenberg-groups
asked Feb 10 at 21:51
Piotr HajlaszPiotr Hajlasz
8,86142969
$endgroup$
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 horizontal distribution in any Carnot group in terms of kernels of some $1$-forms?
I believe the answer should be in the positive and I am looking for references for explicit constructions of such forms so that in the case of the Heisenberg group these constructions would give the contact form.
lie-groups sub-riemannian-geometry heisenberg-groups
lie-groups sub-riemannian-geometry heisenberg-groups
asked Feb 10 at 21:51
Piotr HajlaszPiotr Hajlasz
8,86142969
asked Feb 10 at 21:51
Piotr HajlaszPiotr Hajlasz
8,86142969
asked Feb 10 at 21:51
Piotr HajlaszPiotr Hajlasz
8,86142969
asked Feb 10 at 21:51
Piotr HajlaszPiotr Hajlasz
8,86142969
asked Feb 10 at 21:51
Piotr HajlaszPiotr Hajlasz
8,86142969
8,86142969
add a comment |
add a comment |
1 Answer
1
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oldest
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$begingroup$
The distribution on any Carnot group is left invariant, so pick any set of covectors at the origin that annihilates the distribution at that point, then left translate it to give a global set of one forms whose kernel is the distribution.
On the Heisenberg group this procedures gives a scalar multiple of the standard contact form.
answered Feb 10 at 21:53
RazielRaziel
2,08411425
$endgroup$
$begingroup$
Thank, you that was very simple.
$endgroup$
– Piotr Hajlasz
Feb 10 at 23:13
$begingroup$
At the origin this are just linear forms on the tangent space. You get (left-invariant) differential forms once you translate.
$endgroup$
– YCor
Feb 11 at 0:17
add a comment |
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1 Answer
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1 Answer
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$begingroup$
The distribution on any Carnot group is left invariant, so pick any set of covectors at the origin that annihilates the distribution at that point, then left translate it to give a global set of one forms whose kernel is the distribution.
On the Heisenberg group this procedures gives a scalar multiple of the standard contact form.
answered Feb 10 at 21:53
RazielRaziel
2,08411425
$endgroup$
$begingroup$
Thank, you that was very simple.
$endgroup$
– Piotr Hajlasz
Feb 10 at 23:13
$begingroup$
At the origin this are just linear forms on the tangent space. You get (left-invariant) differential forms once you translate.
$endgroup$
– YCor
Feb 11 at 0:17
add a comment |
$begingroup$
The distribution on any Carnot group is left invariant, so pick any set of covectors at the origin that annihilates the distribution at that point, then left translate it to give a global set of one forms whose kernel is the distribution.
On the Heisenberg group this procedures gives a scalar multiple of the standard contact form.
answered Feb 10 at 21:53
RazielRaziel
2,08411425
$endgroup$
$begingroup$
Thank, you that was very simple.
$endgroup$
– Piotr Hajlasz
Feb 10 at 23:13
$begingroup$
At the origin this are just linear forms on the tangent space. You get (left-invariant) differential forms once you translate.
$endgroup$
– YCor
Feb 11 at 0:17
add a comment |
$begingroup$
The distribution on any Carnot group is left invariant, so pick any set of covectors at the origin that annihilates the distribution at that point, then left translate it to give a global set of one forms whose kernel is the distribution.
On the Heisenberg group this procedures gives a scalar multiple of the standard contact form.
answered Feb 10 at 21:53
RazielRaziel
2,08411425
$endgroup$
The distribution on any Carnot group is left invariant, so pick any set of covectors at the origin that annihilates the distribution at that point, then left translate it to give a global set of one forms whose kernel is the distribution.
On the Heisenberg group this procedures gives a scalar multiple of the standard contact form.
answered Feb 10 at 21:53
RazielRaziel
2,08411425
edited Feb 11 at 8:35
answered Feb 10 at 21:53
RazielRaziel
2,08411425
answered Feb 10 at 21:53
RazielRaziel
2,08411425
answered Feb 10 at 21:53
RazielRaziel
2,08411425
2,08411425
$begingroup$
Thank, you that was very simple.
$endgroup$
– Piotr Hajlasz
Feb 10 at 23:13
$begingroup$
At the origin this are just linear forms on the tangent space. You get (left-invariant) differential forms once you translate.
$endgroup$
– YCor
Feb 11 at 0:17
add a comment |
$begingroup$
Thank, you that was very simple.
$endgroup$
– Piotr Hajlasz
Feb 10 at 23:13
$begingroup$
At the origin this are just linear forms on the tangent space. You get (left-invariant) differential forms once you translate.
$endgroup$
– YCor
Feb 11 at 0:17
$begingroup$
Thank, you that was very simple.
$endgroup$
– Piotr Hajlasz
Feb 10 at 23:13
$begingroup$
Thank, you that was very simple.
$endgroup$
– Piotr Hajlasz
Feb 10 at 23:13
$begingroup$
At the origin this are just linear forms on the tangent space. You get (left-invariant) differential forms once you translate.
$endgroup$
– YCor
Feb 11 at 0:17
$begingroup$
At the origin this are just linear forms on the tangent space. You get (left-invariant) differential forms once you translate.
$endgroup$
– YCor
Feb 11 at 0:17
add a comment |
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