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Difference between revisions of "Distribution of tangent subspaces"

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=== Integrability of distributions ===
 
=== Integrability of distributions ===
<a name=integrability> A distribution $L=\{L_x\}$ of rank $k$,
+
A distribution $L=\{L_x\}$ of rank $k$, $0<k<n$, is called
$0<k<n$, is called integrable in a domain $U\subseteq M$, if
+
integrable in a domain $U\subseteq M$, if through each point
through each point $a\in M$ passes the germ of a
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$a\in M$ passes the germ of a $k$-dimensional submanifold
$k$-dimensional submanifold $N_a=N$ which is tangent to the
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$N_a=N$ which is tangent to the distribution: $\forall x\in N\
distribution: $\forall x\in N\ L_x=T_xN\subset T_xM$.
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L_x=T_xN\subset T_xM$.
  
 
'''Example'''. Each vector field defines an integrable
 
'''Example'''. Each vector field defines an integrable
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$\xi_1,\dots,\xi_k$ is integrable if and only if
 
$\xi_1,\dots,\xi_k$ is integrable if and only if
 
$$
 
$$
\rd \xi_i=\sum_{s=1}^k\eta_s\land\xi_s
+
\rd \xi_i=\sum_{s=1}^k\eta_{is}\land\xi_s
 
$$
 
$$
with suitable 1-forms $\eta_1,\dots,\eta_k$.
+
with suitable 1-forms $\eta_{is}$.

Revision as of 10:26, 3 May 2012

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 $\bigcup_i\operatorname{Ker}\xi_i$ 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: $$ \forall i,j\quad [v_i,v_j]=\sum_{s=1}^k \varphi_{ijs} v_s $$ with suitable 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 $\eta_{is}$.

How to Cite This Entry:
Distribution of tangent subspaces. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Distribution_of_tangent_subspaces&oldid=25884