Building Charts from Vector Fields
Sorry for the delay; I’ve been swamped at my actual job the last couple days.
Anyway, I left off last time by pointing out that coordinate vector fields commute:
Today I want to show a certain converse: if we have vector fields
on some open region
that are linearly independent at some
and which commute —
for all
and
— then we can find some coordinate chart
around
so that the
are the first
coordinate vector fields. That is,
In particular, if is a vector field with
then there is a coordinate chart
around
with
.
If is any chart, then we can describe the
th coordinate vector field by saying it’s the unique vector field on
that is
-related to the
th partial derivative in
. That is, we’re trying to prove that:
on
.
In fact, we can further simplify our claim by assuming that ,
, and
— that the vector fields agree at the point
. Indeed, if
is any coordinate map taking
to
then we can define the vector fields
on
. These must have vanishing brackets because we can calculate:
What’s more, if is a local diffeomorphism of
with
, then
is a coordinate map satisfying our assertion.
Now, let be the flow of
, and let
be a small enough neighborhood of
that we can define
by
The order that the flows come in here doesn’t matter, since we’re assuming that the — and thus their flows — commute. Anyway, given any smooth test function
on
we can check
That is, . To see this for any other
, simply swap around the flows to bring
to the front.
We can also check that
Thus is the identity transformation on
. The inverse function theorem now tells us that there is a chart
around
with
, which will then satisfy our assertions.
[…] we can find a patch and an everywhere-nonzero vector field on so that spans for every . Then we know we can find a chart around such that on . Then the curve with coordinates and for all other […]
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