Distributions Integral Manifolds and the Frobenius Theorem: Examples

The fields ${\mathrm{\partial }}_{x^1},\dots ,{\mathrm{\partial }}_{x^k}$ are smooth and pointwise [given] independent, so they define a smooth rank-$k$ distribution.

For fixed constants $c_{k+1},\dots ,c_n$, the affine plane [given] $\{x^{k+1}=c_{k+1},\dots ,x^n=c_n\}$ has tangent space spanned by those same coordinate fields at every point. Hence it is an integral manifold.

Every connected integral manifold has constant transverse coordinates, so it [given] lies in one of the affine planes from step 1.2. Those full planes are connected and cannot be enlarged while retaining that property. Therefore they are exactly the maximal leaves.

Because the third partial derivative of $F$ is $1$, the map is a [given] submersion everywhere.

The kernel distribution is therefore integrable, and its maximal connected [given] integral manifolds are the connected components of the level sets of $F$. Each level set here is a connected paraboloid.

Thus $\ker dF$ is an explicit integrable distribution. [given] ∎

On the punctured plane, the differential $df$ is never zero, so $f$ is a [given] submersion to $(0,\mathrm{\infty })$.

Therefore $\ker df$ is an integrable line distribution, and its leaves are [given] the connected components of the level sets $x^2+y^2=c$. Those components are exactly the circles of radius $\sqrt{c}$.

The correspondence between integrable distributions and regular foliations [given] turns this family of circles into a regular foliation of ${\mathbb{R}}^2\setminus \{0\}$.

In a product chart, the slices $v=\text{constant}$ are exactly the local [given] pieces of the fibres $M\times \{q\}$. Their connected components are therefore the local plaques.

Transition maps preserve the second coordinate up to a change depending only [given] on the old second coordinate, so these charts satisfy the defining condition of a regular foliation atlas.

Hence the connected components of the fibres of the projection [given] $M\times N\to N$ form the product foliation.

The field is nowhere zero on ${\mathbb{R}}^2\setminus \{0\}$, so it defines a [given] regular one-dimensional distribution there.

Along the flow, [given] $X(x^2+y^2)=2x(-y)+2y(x)=0,$ so the radius is constant on every orbit. The orbits are therefore contained in circles about the origin, and conversely each such circle is an orbit.

Hence the punctured plane is foliated by the rotation circles. At the [given] origin the field vanishes, so the regular rank-one hypothesis fails there.

On the smooth torus of [A2], the constant line field spanned by $(1,\alpha )$ [A2] is smooth and nowhere zero, so it gives a regular one-dimensional distribution on ${\mathbb{T}}^2$.

The integral curve through $[p]$ is the projected affine line [given] $t\mapsto [p+(t,\alpha t)]$. For $p=0$, fix a point $[(u,v)]$ of the torus and a neighborhood of it. Because $\alpha$ is irrational, the set of classes $\{[n\alpha ]:n\in \mathbb{Z}\}$ is dense in $\mathbb{R}\mathrm{/}\mathbb{Z}$. Choose $n$ so that $[n\alpha ]$ is arbitrarily close to $[v-\alpha u]$, and set $t=u+n$. Then $[(t,\alpha t)]=[(u,\alpha u+n\alpha )]$ has first coordinate $[u]$ and second coordinate arbitrarily close to $[v]$. Thus the leaf through $[0]$ is dense. Every other leaf is a torus translate of this one, and translations are homeomorphisms, so every leaf is dense.

Therefore the torus carries a regular foliation with dense, nonembedded [given] leaves.

The quotient map identifies horizontal tangent directions with horizontal [given] tangent directions, so the horizontal line field descends to a smooth one-dimensional distribution on the Mobius band.

Its local plaques are the images of small horizontal intervals, and the [given] quotient charts preserve that plaque structure. Hence the descended line field defines a regular foliation.

The line $t=0$ closes up to the central circle. If $t\mathrm{\ne }0$, then [given] the endpoint $(1,t)$ is identified with $(0,-t)$, so one horizontal pass through the quotient changes the transverse sign; a second pass along the line $-t$ returns to the original class. Thus the leaf through $t\mathrm{\ne }0$ is also a circle.

This yields the standard line foliation of the Mobius band. [given] ∎

The two fields $X_1$ and $X_2$ are smooth and pointwise independent, so [given] they define a smooth rank-$2$ distribution on ${\mathbb{R}}^3$.

Their bracket is [given] $[X_1,X_2]=-{\mathrm{\partial }}_z,$ which is not a linear combination of $X_1$ and $X_2$. Therefore the distribution is not involutive, even on this global frame.

By [L1], an integrable distribution would have to be involutive. Hence this standard [L1] contact distribution is a counterexample to the claim that it is integrable.

Therefore the displayed statement is refuted. [given] ∎

Any smooth vector field tangent to this family must vanish at the origin. [given] The Lie bracket of two such fields also vanishes at the origin, so the tangent fields are closed under bracket in this loose sense.

Nevertheless the family is not a smooth distribution of constant rank, so [given] it does not satisfy the hypotheses of the regular Frobenius theorem.

Hence variable-rank bracket closure does not place a family inside regular [given] Frobenius theory.

The map $t\mapsto I+t{E}_{12}$ is a one-parameter subgroup because [given] $E_{12}^2=0$, so $(I+s{E}_{12})(I+t{E}_{12})=I+(s+t)E_{12}\mathrm{.}$ Hence $H$ is a connected immersed Lie subgroup with tangent space $T_{I}H=\mathfrak{h}$.

For any $g\in G$, the left coset $gH$ has tangent line [given] $d(L_g{)}_I(\mathfrak{h})$ at the point $g$. Thus each coset is an integral manifold of the distribution $\mathcal{D}$.

For each $g\in G$, the curve $c_g(t)=g(I+t{E}_{12})$ has image $gH$ and [given] derivative $c_g^{\mathrm{\prime }}(t)=g{E}_{12}=d(L_{c_g(t)}{)}_I(E_{12}),$ so it is an integral curve of the nowhere-zero rank-$1$ distribution $\mathcal{D}$. Hence any connected integral manifold through $g$ is locally an open piece of that same integral curve and is contained in $gH$. Since step 1.2 shows that $gH$ itself is a connected integral manifold through $g$, [L1] identifies $gH$ as the leaf through $g$. Varying $g$ gives all leaves.