Consecutive shifts along orbits of vector fields
Abstract
Let
M
be a smooth (
C
∞
) manifold,
F
1
,...,
F
n
be vector fields on
M
generating the corresponding flows
Φ
1
,...,
Φ
n
, and
α
1
,...,
α
n
:M→R
smooth functions. Define the following map
f:M→M
by
f(x)=
Φ
n
(...(
Φ
2
(
Φ
1
(x,
α
1
(x)),
α
2
(x)),...,
α
n
(x)).
In this note we give a necessary and sufficient condition on vector fields
F
1
,...,
F
n
and smooth functions
α
1
,...,
α
n
for
f
to be a local diffeomorphism. It turns out that this condition is invariant with respect to the simultaneous permutation of the corresponding vector fields and functions.