Loading…
Homotopically invisible singular curves
Given a smooth manifold M and a totally nonholonomic distribution Δ ⊂ T M of rank d ≥ 3 , we study the effect of singular curves on the topology of the space of horizontal paths joining two points on M . Singular curves are critical points of the endpoint map F : γ ↦ γ ( 1 ) defined on the space Ω o...
Saved in:
Published in: | Calculus of variations and partial differential equations 2017-08, Vol.56 (4), p.1-34, Article 105 |
---|---|
Main Authors: | , , |
Format: | Article |
Language: | English |
Subjects: | |
Citations: | Items that this one cites Items that cite this one |
Online Access: | Get full text |
Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
Summary: | Given a smooth manifold
M
and a totally nonholonomic distribution
Δ
⊂
T
M
of rank
d
≥
3
, we study the effect of singular curves on the topology of the space of horizontal paths joining two points on
M
. Singular curves are critical points of the endpoint map
F
:
γ
↦
γ
(
1
)
defined on the space
Ω
of horizontal paths starting at a fixed point
x
. We consider a sub-Riemannian energy
J
:
Ω
(
y
)
→
R
, where
Ω
(
y
)
=
F
-
1
(
y
)
is the space of horizontal paths connecting
x
with
y
, and study those singular paths that do not influence the homotopy type of the Lebesgue sets
{
γ
∈
Ω
(
y
)
|
J
(
γ
)
≤
E
}
. We call them
homotopically invisible
. It turns out that for
d
≥
3
generic sub-Riemannian structures in the sense of Chitour et al. (J Differ Geom 73(1):45–73,
2006
) have only homotopically invisible singular curves. Our results can be seen as a first step for developing the calculus of variations on the singular space of horizontal curves (in this direction we prove a sub-Riemannian minimax principle and discuss some applications). |
---|---|
ISSN: | 0944-2669 1432-0835 |
DOI: | 10.1007/s00526-017-1203-z |