Vector Fields, Flows, and the Lie Derivative

A vector field assigns a tangent vector to each point of a manifold — an infinitesimal flow. Integrating it gives a one-parameter family of diffeomorphisms; differentiating tensors along it gives the Lie derivative; and the failure of two flows to commute is measured by the Lie bracket, which governs when a family of vector fields can be integrated to a submanifold (Frobenius). This page extends the tangent-space material of topology-manifolds.md §3 and opens the Differential Geometry folder.

Vector fields

A vector field $X$ on a smooth manifold $M$ is a smooth assignment $p\mapsto X_p\in T_{p}M$ — a smooth section of the tangent bundle $TM$ (see fiber-bundles.md for the bundle language). In local coordinates $x^i$, $$X=X^i(x)\frac{\partial }{\partial x^i},$$ with smooth components $X^i$. Equivalently, $X$ is a derivation of the algebra $C^{\infty }(M)$: a linear map $f\mapsto X(f)=X^i{\partial }_{i}f$ satisfying the Leibniz rule $X(fg)=fX(g)+gX(f)$. The space of vector fields is written $\mathfrak{X}(M)$.

Flows

A vector field is the velocity field of a flow. An integral curve through $p$ is a curve $\gamma$ with $\gamma (0)=p$ and $\dot{\gamma }(t)=X_{\gamma (t)}$ — in coordinates, the ODE system ${\dot{x}}^i(t)=X^i(x(t))$. By existence–uniqueness of ODEs these integral curves fit together into the flow ${\Phi }_t$, a one-parameter (local) group of diffeomorphisms: $${\Phi }_0=\mathrm{id},{\Phi }_{s+t}={\Phi }_s\circ {\Phi }_t,\frac{d}{dt}{\Phi }_t(p)=X_{{\Phi }_t(p)}.$$ If every integral curve extends to all $t\in \mathbb{R}$ the field is complete (automatic on compact $M$). The flow is the finite object; the vector field is its infinitesimal generator — exactly the exponential-map relationship, here on the infinite-dimensional diffeomorphism group.

The Lie bracket

Two vector fields $X,Y$ generate flows that generally fail to commute. The leading-order failure is the Lie bracket $$[X,Y]=XY-YX,[X,Y{]}^k=X^i{\partial }_i{Y}^k-Y^i{\partial }_i{X}^k,$$ again a derivation, hence a vector field. It is bilinear, antisymmetric, and satisfies the Jacobi identity, so $(\mathfrak{X}(M),[\cdot ,\cdot ])$ is an (infinite-dimensional) Lie algebra — the abstract structure of lie-algebras.md. Geometrically, $[X,Y]$ measures the gap when one flows along $X$ for time $\sqrt{t}$, then $Y$, then back along $X$, then back along $Y$: the loop fails to close by $t[X,Y]+O(t^{3/2})$.

The Lie derivative

The flow of $X$ lets one transport and hence differentiate any tensor field. The Lie derivative ${\mathcal{L}}_{X}T$ is the infinitesimal rate of change of a tensor $T$ dragged along the flow: $${\mathcal{L}}_{X}T=\frac{d}{dt}{|}_{t=0}{\Phi }_t^{\ast }T.$$ Special cases:

• On functions, ${\mathcal{L}}_{X}f=X(f)$.
• On vector fields, ${\mathcal{L}}_{X}Y=[X,Y]$.
• On forms, Cartan's magic formula ${\mathcal{L}}_X=d{\iota }_X+{\iota }_{X}d$, where ${\iota }_X$ is contraction with $X$ (see tensors-and-forms.md).

Unlike the covariant derivative of connections.md, the Lie derivative needs no extra structure — only the flow. A tensor is invariant under the symmetry generated by $X$ iff ${\mathcal{L}}_{X}T=0$; for the metric this defines a Killing vector (a continuous isometry — see geometry/isometry-groups.md).

The Frobenius theorem

When can a family of vector fields be "integrated" to a family of submanifolds? A rank-$k$ distribution $\mathcal{D}$ assigns a $k$-dimensional subspace ${\mathcal{D}}_p\subset T_{p}M$ smoothly. It is involutive if $X,Y\in \mathcal{D}\Rightarrow [X,Y]\in \mathcal{D}$, and integrable if through each point there is a submanifold everywhere tangent to $\mathcal{D}$.

Frobenius theorem. A smooth distribution is integrable iff it is involutive.

So the Lie bracket is the exact obstruction to integrability. This is the geometric heart of many "consistency conditions" in physics, and the finite-dimensional shadow of the Lie-group ↔ Lie-subalgebra correspondence. It reappears as the integrability / flatness condition for connections in curvature.md.

References

• Lee, Introduction to Smooth Manifolds, Ch. 8–9, 19.
• Warner, Foundations of Differentiable Manifolds and Lie Groups, Ch. 1.
• Spivak, A Comprehensive Introduction to Differential Geometry, Vol. 1.