Distributions
A vector field 
defines a one-dimensional subspace of 
at any point 
with 
: the subspace spanned by 
. If is everywhere nonzero, then it defines a one-dimensional subspace of each tangent space. A distribution generalizes this sort of thing to higher dimensions.
To this end, we define a 
-dimensional distribution 
on an 
-dimensional manifold 
to be a map 
, where is a -dimensional subspace of . Further, we require that this map be “smooth”, in the sense that for any 
there exists some neighborhood 
of 
and vector fields 
such that the vectors 
span 
for each 
.
Notice here that the 
don’t have to work for the whole manifold . Indeed, we will see that in many cases there are no everywhere-nonzero vector fields on a manifold . But over a small patch we might more easily find vector fields that are linearly independent at each point, and thus define a smooth -dimensional distribution over . Then more general smooth distributions come from patching these sorts of smooth distributions together.
A vector field 
on “belongs to” a distribution — which we write 
— if 
for all 
. We say that is “integrable” if ![[X,Y]\in\Delta](http://s0.wp.com/latex.php?latex=%5BX%2CY%5D%5Cin%5CDelta&bg=e6e6e6&fg=333333&s=0 "[X,Y]\in\Delta")
for all and 
belonging to .
Every one-dimensional manifold is integrable. To see this, we note that if and belong to then 
for some constant 
, at least at those points where 
. Thus we see that
![\displaystyle[X,Y]=[fY,Y]=f[Y,Y]-(Yf)Y=-(Yf)Y](http://s0.wp.com/latex.php?latex=%5Cdisplaystyle%5BX%2CY%5D%3D%5BfY%2CY%5D%3Df%5BY%2CY%5D-%28Yf%29Y%3D-%28Yf%29Y&bg=e6e6e6&fg=333333&s=0 "\displaystyle[X,Y]=[fY,Y]=f[Y,Y]-(Yf)Y=-(Yf)Y")
and so ![[X,Y]](http://s0.wp.com/latex.php?latex=%5BX%2CY%5D&bg=e6e6e6&fg=333333&s=0 "[X,Y]")
is proportional to , and thus belongs to . To handle points where 
, we can put the scalar multiplier on the other side.
Like this:
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June 28, 2011 -
Posted by John Armstrong |
Differential Topology, Topology
I’m not understanding the last equation [fY,Y] = f[Y,Y] – (Y f)Y — could you elaborate?
Thanks!
Well, let’s work it out:
Now just use
for both 
and 
.
OK, I get it now, thanks.
I think my difficulty comes from confusion over when juxtaposition denotes multiplication in the ring, multiplication of the ring over the module, function application, function composition, or even something else. (I had similar difficulties with representation theory
In a term like “fXYg”, the empty spaces between the letters can mean different things depending on how it’s parenthesized; I still haven’t wrapped my head around whether all the different ways make sense and that (among the ones that do make sense — all of them?) they all mean the same thing.
[...] a -dimensional distribution on an -dimensional manifold , we say that a -dimensional submanifold is an “integral [...]
Pingback by Integral Submanifolds « The Unapologetic Mathematician | June 30, 2011 |
It’s possible to fully-parenthesize these sorts of expressions, but then they get confusing for a different reason. In general, I try to write things out so they associate to the right, because of function application, and use parentheses for particularly ambiguous setups.
In
, we could write 
, but we’ll conventionally drop the parens and write 
for the application of the vector field to the function. There’s no way of “multiplying” vector fields, so 
has to be interpreted as 
. And it doesn’t matter whether we multiply 
by 
before applying it to 
or after, since 
and ](http://s0.wp.com/latex.php?latex=%5BfX%5D%28Yg%29&bg=e6e6e6&fg=333333&s=0 "[fX](Yg)")
give the same result.
[...] say that we have a one-dimensional distribution on a manifold . Around any point we can find a patch and an everywhere-nonzero vector field on [...]
Pingback by Integrable Distributions Have Integral Submanifolds « The Unapologetic Mathematician | June 30, 2011 |
[...] of the foliation. We also ask that the tangent spaces to the leaves define a -dimensional distribution on , which we say is “induced” by , and that any connected integral submanifold of [...]
Pingback by Foliations « The Unapologetic Mathematician | July 1, 2011 |