Distribution (differential geometry)

A rank-$k$ distribution on a smooth manifold is a rank-$k$ vector subbundle of its tangent bundle. It is involutive when the Lie bracket of vector fields of any two local sections is again a section.

For a codimension-one tangent distribution, a coorientation is an orientation of its normal quotient line bundle. An oriented real line bundle is trivial: local positive sections can be combined with a partition of unity without cancellation. Hence a cooriented distribution is the kernel of a global nowhere-vanishing one-form. This choice is independent of the orientation of the ambient manifold, and reversing it does not change the Godbillon-Vey invariant.

A smooth distribution is involutive if its local smooth sections are closed under the Lie bracket of vector fields. The Frobenius theorem identifies this algebraic condition with being an integrable distribution.

A distribution is integrable when every point lies on an immersed submanifold whose tangent spaces equal the distribution. Such submanifolds are its integral manifolds.

An integral manifold of a rank-$r$ smooth distribution $D$ is an immersed $r$-dimensional submanifold with $T_{p}N=D_p$ at every point. For an integrable distribution, maximal connected integral manifolds form its leaves.

The map descends to an injective immersion from $\mathbb{R}\times (\mathbb{R}/\mathbb{Z})$ into the three-dimensional torus. Injectivity follows because an integer difference in $s$ with an integer difference in $\alpha s$ must be zero. Its image is dense by an irrational rotation of the circle, and is tangent to the commuting independent fields ${\partial }_x+\alpha {\partial }_z$ and ${\partial }_y$. It is a leaf of a regular foliation, but cannot be an embedded submanifold: in an embedding chart the coordinate plane would be locally closed yet contain a dense ambient subset.

A smooth curve tangent to a regular integrable distribution lies in one maximal leaf of a regular foliation. In a Frobenius theorem chart its transverse coordinates have zero derivative, so each short segment lies in one plaque. A finite chain of such segments covers any compact parameter interval and joins plaques in the same leaf. A single embedded local plaque only gives confinement while the curve remains in its chart. A dense immersed cylinder in a three-dimensional torus shows why global embedded confinement cannot generally replace the immersed statement.

A constant-rank smooth distribution $D$ is an integrable distribution exactly when it is involutive. In differential-form language, let ${\alpha }^1,\dots ,{\alpha }^q$ be a local basis of its annihilator, so $D={\bigcap }_i\ker {\alpha }^i$. The equivalent condition isfor suitable local one-forms ${\beta }^i{}_j$. More precisely, the ideal generated by the annihilating forms inside the exterior algebra of differential forms is closed under the exterior derivative; this is an exterior differential ideal. It is not a claim that a two-form lies in the one-form annihilator itself.

For local vector fields $X,Y$ tangent to $D$, the definition of exterior derivative givesThus the displayed closure condition says exactly that the Lie bracket of vector fields remains tangent to $D$. The theorem turns this local algebraic condition into integral manifolds through every point.

For a nowhere-zero one-form $\alpha$, its kernel is tangent to local hypersurfaces precisely when $\alpha \wedge d\alpha =0$. In general relativity, applying this to a metric-dual one-form characterizes hypersurface orthogonality and vanishing twist.

An exterior differential ideal on a smooth manifold is a graded ideal $\mathcal{I}$ in the exterior algebra ${\Omega }^{\ast }(M)$ of differential forms that is closed under the exterior derivative:Being an ideal also requires $\alpha \wedge \beta \in \mathcal{I}$ for every $\beta \in {\Omega }^{\ast }(M)$. Thus stability under differentiation is an additional condition, not a consequence of wedge closure.

For example, the ideal generated by $dz$ in ${\mathbb{R}}^3$ is differential because $d(fdz)=df\wedge dz$. Its integral surfaces include the planes $z=\text{constant}$. In contrast, the ideal generated by $\alpha =dz-xdy$ is not differential: $d\alpha =-dx\wedge dy$ and $\alpha \wedge d\alpha =-dz\wedge dx\wedge dy\ne 0$, whereas wedging $\alpha$ with any element of its generated ideal gives zero. The Frobenius theorem interprets this obstruction as failure of the associated tangent-plane distribution to be integrable.

A nowhere-zero differential one-form $n$ is hypersurface orthogonal if locally $n=Fdt$ with $F\ne 0$. Its kernel consists of tangent vectors to the level hypersurfaces of $t$. Frobenius theorem makes this equivalent to $n\wedge dn=0$, where $d$ is the exterior derivative. With a torsion-free Levi-Civita connection, this is $n_{[\alpha }{\nabla }_{\beta }{n}_{\gamma ]}=0$. For a timelike unit normal, projecting the derivative on both indices gives zero antisymmetric part, which explains why hypersurface orthogonality implies symmetric extrinsic curvature.

For a non-null Killing vector field $V$ of a Levi-Civita connection, hypersurface orthogonality implies that $\eta =V^{♭}/F$, with $F=g(V,V)$, is a closed differential form. The Killing equation and $V^{♭}\wedge d{V}^{♭}=0$ give $2F{\nabla }_{\mu }{V}_{\nu }=V_{\nu }{\partial }_{\mu }F-V_{\mu }{\partial }_{\nu }F$, whence $d\eta =0$. The Poincaré lemma gives $\eta =d\varphi$ locally. For a timelike $V$, $\varphi$ supplies local static time coordinates. This does not assert global exactness.

For an involutive distribution of constant rank $r$, induct on $r$. Straighten a nonzero section by the flow-box theorem, and choose a frame ${\partial }_1,Y_2,\dots ,Y_r$ with no ${\partial }_1$ component in the remaining fields. Involutivity implies ${\partial }_{1}Y=AY$ for their column $Y$. The invertible solution of ${\partial }_{1}B=-BA$, initialized by $B=I$ on $x_1=0$, makes $Z=BY$ independent of $x_1$. Its restriction to the transverse slice is involutive of rank $r-1$, so induction provides coordinates spanning it. Extending these coordinates independently of $x_1$ gives $D=\mathrm{span}({\partial }_1,\dots ,{\partial }_r)$. Conversely, fields tangent to an integral manifold have tangent brackets because they preserve the ideal of smooth functions vanishing on it.

A plane distribution on a three-manifold is a rank-two distribution. Locally it is the kernel of a nowhere-zero differential 1-form $\alpha$.

The Frobenius theorem gives the local criterion

A contact structure on a three-manifold is a plane distribution locally of the form $\ker \alpha$ with $\alpha \wedge d\alpha$ nowhere zero. It is maximally nonintegrable; the standard example on ${\mathbb{R}}^3$ is $\ker (dz-xdy)$.

A smooth interval family of contact forms on a compact manifold without boundary is carried back to its initial contact distribution by a smooth isotopy. The pullback of each form is a positive smooth multiple of the initial form. Solve for the horizontal generator of contact stability and integrate the resulting scalar equation for the contact conformal factor along an isotopy.

If a time-dependent vector field satisfies ${\dot{\alpha }}_t+{\mathcal{L}}_{Y_t}{\alpha }_t=h_t{\alpha }_t$, its flow maps satisfy ${\rho }_t^{\ast }{\alpha }_t=u_t{\alpha }_0$, where $u_t(x)=\exp ({\int }_0^t{h}_s({\rho }_s(x))ds)$. Thus the conformal factor is smooth and positive even when $h_t$ changes sign.

For a family of contact forms, require $Y_t\in \ker {\alpha }_t$ and solve $({\iota }_{Y_t}d{\alpha }_t){|}_{\ker {\alpha }_t}=-{\dot{\alpha }}_t{|}_{\ker {\alpha }_t}$. The contact distribution symplectic form gives a unique smooth solution. Then ${\dot{\alpha }}_t+{\mathcal{L}}_{Y_t}{\alpha }_t=h_t{\alpha }_t$, with $h_t={\dot{\alpha }}_t(R_t)$ and $R_t$ the Reeb vector field.

A contact form $\alpha$ defines the hyperplane distribution $\xi =\ker \alpha$. Multiplying $\alpha$ by a nowhere-zero function leaves this distribution unchanged. The contact distribution symplectic form distinguishes it from an integrable hyperplane distribution.

For a contact form on a $(2r+1)$-manifold, $d\alpha {|}_{\ker \alpha }$ is nondegenerate: the contact condition says its top exterior power is nowhere zero on the hyperplanes. This fiberwise symplectic form identifies contact-plane vectors with covectors on that plane and determines the horizontal generator of contact stability.

A contact form on a $(2n+1)$-manifold is a one-form satisfying $\alpha \wedge (d\alpha {)}^n\ne 0$ everywhere.

The Reeb vector field of a contact form is uniquely characterized by

For ${\alpha }_s=\frac{1}{2}(r_1^{2}d{\theta }_1+s{r}_2^{2}d{\theta }_2)$ on $S^3$, the Reeb field is $2{\partial }_{{\theta }_1}+2s^{-1}{\partial }_{{\theta }_2}$. If $s$ is irrational, its only closed orbits are the two coordinate circles.