Tangent and Cotangent Bundles - Applications

The Lie derivative provides a coordinate-free way to differentiate tensor fields along vector fields. It captures how geometric quantities change as they flow along integral curves.

DefinitionLie Derivative

Let $X$ be a vector field with flow $\phi_t$ . The Lie derivative of a tensor field $T$ with respect to $X$ is

\mathcal{L}_X T = \lim_{t \to 0} \frac{\phi_t^* T - T}{t}

where $\phi_t^*$ denotes pullback by $\phi_t$ (extended to tensors).

TheoremProperties of Lie Derivative

The Lie derivative satisfies:

$\mathcal{L}_X(Y) = [X, Y]$ for vector fields $Y$
$\mathcal{L}_X(f) = X(f)$ for functions $f$
$\mathcal{L}_X(\omega \wedge \eta) = \mathcal{L}_X\omega \wedge \eta + \omega \wedge \mathcal{L}_X\eta$ (Leibniz rule)
$\mathcal{L}_X(d\omega) = d(\mathcal{L}_X\omega)$ (commutes with exterior derivative)
$\mathcal{L}_{[X,Y]} = [\mathcal{L}_X, \mathcal{L}_Y]$ (Lie algebra homomorphism)

These properties make the Lie derivative the natural differentiation operator in differential geometry, compatible with all geometric structures.

DefinitionCartan's Formula

For differential forms, the Lie derivative is related to the exterior derivative and interior product by

\mathcal{L}_X = i_X \circ d + d \circ i_X

where $(i_X\omega)(Y_1, \ldots, Y_{k-1}) = \omega(X, Y_1, \ldots, Y_{k-1})$ is the interior product.

ExampleLie Derivative on Functions

For $f \in C^\infty(M)$ and vector field $X$ , we have $\mathcal{L}_X f = X(f) = df(X)$ . This shows the Lie derivative generalizes directional derivatives: it measures the rate of change of $f$ along the flow of $X$ .

TheoremSymmetries and Killing Fields

A vector field $X$ generates a symmetry of a Riemannian metric $g$ if and only if $\mathcal{L}_X g = 0$ . Such vector fields are called Killing vector fields. For a function $f$ , $\mathcal{L}_X f = 0$ if and only if $f$ is constant along integral curves of $X$ .

Remark

Killing vector fields are the infinitesimal isometries of a Riemannian manifold. Their study connects differential geometry with Lie group actions and conservation laws in physics.

ExampleRotational Symmetry

On $\mathbb{R}^2$ with the standard metric, the vector field $X = -y\frac{\partial}{\partial x} + x\frac{\partial}{\partial y}$ is a Killing field. Its flow is rotation about the origin, and $\mathcal{L}_X g = 0$ reflects rotational symmetry.

TheoremIntegration of Vector Fields

A vector field $X$ on a compact manifold $M$ integrates to a global flow (one-parameter group of diffeomorphisms $\phi_t: M \to M$ for all $t \in \mathbb{R}$ ) if and only if $M$ is compact. On non-compact manifolds, complete vector fields are those whose integral curves exist for all time.

The distinction between local and global flows is crucial in applications to dynamical systems and Hamiltonian mechanics.