Distribution of tangent subspaces

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 $\Lambda^\bullet(M)=\Lambda^1(M)\oplus\cdots\oplus\Lambda^n(M)$. The involutivity means that this ideal is closed by the exterior differential $\rd:\Lambda^i(M)\to\Lambda^{i+1}(M)$.