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Dimension Collection Point Neighborhood Manifold Subspace Topology Differntiable Function Vector Field Linear Independence Linear Combination Tangent Space Derived Subgroup Span Distribution
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Distribution
In the following we will mean 
when we say smooth.
Definition 1
Let 
be a smooth manifold
of dimension
. Let 
and for each 
, we assign an 
-dimensional subspace
of the tangent space
in such a way that for a
neighbourhood
of 
there exist 
linearly independent
smooth vector fields
such that for any point
,
span
. We let 
refer to the
collection
of all the 
for all 
and we then call 
a
distribution of dimension 
on 
, or sometimes a
-plane distribution on 
. The set of smooth
vector fields
is called a local basis of 
.
be a smooth manifold
of dimension
. Let and for each , we assign an -dimensional subspace
of the tangent space
in such a way that for a
neighbourhood
of there exist linearly independent
smooth vector fields
such that for any point
,
span
. We let refer to the
collection
of all the for all and we then call a
distribution of dimension on , or sometimes a
-plane distribution on . The set of smooth
vector fields
is called a local basis of .Note: The naming is unfortunate here as these distributions have nothing to do with distributions in the sense of analysis. However the naming is in wide use.
Definition 2
We say that a distribution 
on 
is involutive if for every point 
there exists a local basis
in a neighbourhood of 
such that for all
, ![$ [X_i,X_j]$](http://myyn.org/static/assets/mathbrowser/article/397/images/img32.png)
(the commutator
of two vector fields) is in the span of
. That is, if
![$ [X_i,X_j]$](http://myyn.org/static/assets/mathbrowser/article/397/images/img34.png)
is a linear combination
of
.
Normally this is written as
![$ [ \Delta , \Delta ] \subset \Delta$](http://myyn.org/static/assets/mathbrowser/article/397/images/img36.png)
.
on
is involutive if for every point there exists a local basis
in a neighbourhood of such that for all
,
(the commutator
of two vector fields) is in the span of
. That is, if
is a linear combination
of
.
Normally this is written as
.
Bibliography
- 1
- William M. Boothby. An Introduction to Differentiable Manifolds and Riemannian Geometry, Academic Press, San Diego, California, 2003.