Tangent and Cotangent Bundles - Main Theorem

Frobenius' theorem characterizes when a distribution of tangent spaces integrates to give a foliation. It provides the fundamental criterion for solving systems of partial differential equations geometrically.

TheoremFrobenius Theorem

Let $\mathcal{D}$ be a $k$ -dimensional distribution on $M$ (a smooth assignment of $k$ -dimensional subspaces $\mathcal{D}_p \subset T_pM$ ). The following are equivalent:

$\mathcal{D}$ is involutive: if $X, Y$ are vector fields tangent to $\mathcal{D}$ , then $[X, Y]$ is also tangent to $\mathcal{D}$
$\mathcal{D}$ is integrable: every point has a neighborhood $U$ with a chart $(U, \varphi)$ such that the last $n-k$ coordinates are constant on each integral manifold
Through each point passes a unique maximal connected integral manifold

This theorem bridges algebra (the Lie bracket condition) and geometry (existence of integral submanifolds). It's essential for understanding when local symmetries extend to global geometric structures.

Remark

The involutivity condition $[X, Y] \in \mathcal{D}$ for all $X, Y \in \mathcal{D}$ is algebraic and checkable, while integrability is a geometric property. Frobenius' theorem shows these are equivalent, providing a practical test for existence of foliations.

ExampleNon-integrable Distribution

On $\mathbb{R}^3$ with coordinates $(x, y, z)$ , consider the distribution spanned by

X_1 = \frac{\partial}{\partial x} + y\frac{\partial}{\partial z}, \quad X_2 = \frac{\partial}{\partial y}

Then $[X_1, X_2] = \frac{\partial}{\partial z}$ , which is not in the span of $X_1, X_2$ . Thus this distribution is not involutive and cannot be integrated.

TheoremTensor Characterization

A distribution $\mathcal{D}$ is involutive if and only if for every covector field $\omega$ annihilating $\mathcal{D}$ (meaning $\omega(X) = 0$ for all $X \in \mathcal{D}$ ), we have

d\omega \wedge \omega = 0

This is equivalent to the exterior derivative $d\omega$ also annihilating $\mathcal{D}$ when restricted appropriately.

DefinitionFoliation

A foliation of dimension $k$ on $M$ is a decomposition of $M$ into disjoint connected submanifolds (called leaves) of dimension $k$ , such that locally the foliation looks like the decomposition of $\mathbb{R}^n$ into $\mathbb{R}^k \times \{*\}$ slices.

ExampleFoliation of the Torus

The 2-torus $T^2 = S^1 \times S^1$ admits a 1-dimensional foliation by curves $\{(e^{i\theta}, e^{i\alpha\theta}) : \theta \in [0, 2\pi)\}$ for each slope $\alpha$ . When $\alpha$ is rational, the leaves are closed circles. When $\alpha$ is irrational, each leaf is dense in $T^2$ .

TheoremLie's Third Theorem (Local Version)

Every finite-dimensional Lie algebra $\mathfrak{g}$ arises as the Lie algebra of vector fields generating a local Lie group action on some manifold.

This deep result shows that abstract Lie algebras have geometric realizations, connecting representation theory with differential geometry.