View source for Distribution of tangent subspaces

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''Distribution'' on a smooth manifold is a collection of subspaces $L_a\subseteq T_aM$ of the tangent spaces $T_a M$, which depends in a regular way (smooth, analytic etc.) on the ''base point'' $a\in M$. In particular, the dimension of all subspaces should be constant (the ''dimension'', sometimes the ''rank'' of the distribution). If the regularity fails on a small subset $\varSigma\subset M$, one sometimes says about ''singular distribution'' with the ''singular locus'' $\varSigma$.

One-dimensional distributions with $\dim L_a=1$ are sometimes called the ''line fields''.

=== Definitions ===
In formal terms, a distribution is a subset of the tangent bundle $TM$, which itself has the inherited structure of the vector bundle over $M$. Usually the cases of $0$-dimensional and $n$-dimensional subspaces are excluded from consideration.

If $v_1,\dots,v_k$ are vector fields on $M$, their span is a distribution provided that the rank of the tuple of fields is constant over all points of $m$. A single vector field $v$ defines a line field (distribution of rank 1) over the set of points $M\smallsetminus\varSigma$ of its nonzero values, where $\varSigma=\{a\in M:\ v(a)=0\}$.

If $\xi_1,\dots,\xi_l\in\Lambda^1(M)$ are differential 1-forms on $M$, then their common null spaces $\bigcap_i\operatorname{Ker}\xi_i$ is a distribution provided that the rank of the tuple of forms is constant over all points of $M$.

=== Integrability of distributions ===
A distribution $L=\{L_x\}$ of rank $k$, $0<k<n$, is called
integrable in a domain $U\subseteq M$, if through each point
$a\in M$ passes the germ of a $k$-dimensional submanifold
$N_a=N$ which is tangent to the distribution: $\forall x\in N\
L_x=T_xN\subset T_xM$.

'''Example'''. Each vector field defines an integrable
distribution outside of its zero locus: the 1-dimensional
submanifold (curve) through a point $a$ is the integral curve
of $v$ with the initial condition at this point.

'''Frobenius integrability theorem'''. A distribution spanned
by a tuple of vector fields $v_1,\dots,v_k$ over the set where
their rank is $k$, is integrable if and only if their
commutators belong to the span:
$$
\forall i,j=1,\dots,k\qquad [v_i,v_j]=\sum_{s=1}^k
\varphi_{ijs} v_s
$$
with suitable smooth functions $\varphi_{ijs}$.

A distribution spanned by a tuple of $1$-forms
$\xi_1,\dots,\xi_k$ is integrable if and only if
$$
\rd \xi_i=\sum_{s=1}^k\eta_{is}\land\xi_s
$$
with suitable 1-forms $\eta_{is}$.

====Comments====
* The conditions appearing in the Frobenius theorem, are often referred to as ''involutivity'' of the distribution.
* The involutivity is automatic for 1-dimensional distributions: this is reflected in absence of extra integrability conditions in the theorem of existence of solutions of ordinary differential equations.
* An involutive system of vector fields can always be locally generated by ''commuting'' vector fields $w_1,\dots,w_k$ with $[w_i,w_j]=0$ for all $i,j$.
* Algebraically, a distribution cannot be seen as a finitely generated $C^\infty(M)$-submodule of sections of the tangent bundle $TM$ (generated by the fields $v_1,\dots,v_k$). The involutivity means that this submodule is closed by the commutator action.
* Algebraically, a distribution can be seen as an ideal (generated by the forms $\xi_1,\dots,\xi_k)$ in the graded  [[exterior algebra]] $\Lambda^\bullet(M)=\Lambda^0(M)\oplus\Lambda^1(M)\oplus\cdots\oplus\Lambda^n(M)$. The involutivity means that this ideal is closed by the [[exterior differential]] $\rd:\Lambda^i(M)\to\Lambda^{i+1}(M)$.

=== References ===
{|
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|valign="top"|{{Ref|B}}||valign="top"|Boothby, W. M. ''An introduction to differentiable manifolds and Riemannian geometry'', Pure and Applied Mathematics, '''120'''. Academic Press, Inc., Orlando, FL, 1986, {{MR|0861409}}	
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|valign="top"|{{Ref|W}}||valign="top"|Warner, F. W. ''Foundations of differentiable manifolds and Lie groups''. Graduate Texts in Mathematics, '''94'''. Springer-Verlag, New York-Berlin, 1983. {{MR|0722297}}
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|valign="top"|{{Ref|M}}||valign="top"|Morita, S. ''Geometry of differential forms'', Translations of Mathematical Monographs, '''201'''. Iwanami Series in Modern Mathematics. American Mathematical Society, Providence, RI, 2001. {{MR|1851352}}
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