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 $\rd:\Lambda^i(M)\to\Lambda^{i+1}(M)$.

References

Boothby, William M. An introduction to differentiable manifolds and Riemannian geometry, Pure and Applied Mathematics, 120. Academic Press, Inc., Orlando, FL, 1986, MR0861409

Warner, F. W. Foundations of differentiable manifolds and Lie groups. Graduate Texts in Mathematics, 94. Springer-Verlag, New York-Berlin, 1983. MR0722297

Morita, S. Geometry of differential forms, Translations of Mathematical Monographs, 201. Iwanami Series in Modern Mathematics. American Mathematical Society, Providence, RI, 2001. MR1851352