The Lie derivative differentiates geometric objects along a flow. Unlike a connection, it needs no extra structure; it measures change under the diffeomorphisms generated by a vector field.
Definition by flow
Definition. If \(X\) has local flow \(\Phi_t\), the Lie derivative of a covariant tensor \(T\) is \[ \mathcal L_XT=\left.\frac{\mathrm{d}}{\mathrm{d}t}\right|_{t=0}\Phi_t^*T. \] For a vector field \(Y\), it is \[ \mathcal L_XY=\left.\frac{\mathrm{d}}{\mathrm{d}t}\right|_{t=0} (\Phi_{-t})_*Y. \]
Proposition. \[ \mathcal L_Xf=Xf,\qquad \mathcal L_XY=[X,Y], \] and \(\mathcal L_X\) is a derivation for tensor products and contractions.
In coordinates, for a one-form \(\alpha\), \[ (\mathcal L_X\alpha)_i=X^j\partial_j\alpha_i+\alpha_j\partial_iX^j. \] For a metric \(g\), \[ (\mathcal L_Xg)_{ij} =X^k\partial_kg_{ij}+g_{kj}\partial_iX^k+g_{ik}\partial_jX^k. \]
Interior multiplication
Definition. For \(\omega\in\Omega^k(M)\), contraction with \(X\) is \[ (\iota_X\omega)(X_2,\ldots,X_k) =\omega(X,X_2,\ldots,X_k). \]
Theorem (Cartan's magic formula). For every differential form, \[ \mathcal L_X\omega=\mathrm{d}(\iota_X\omega)+\iota_X(\mathrm{d}\omega). \]
Proof. Both sides are degree-zero derivations of the exterior algebra and commute with pullback. It is enough to check functions and their differentials, which locally generate forms. For \(f\), \[ \mathrm{d}\iota_Xf+\iota_X\mathrm{d}f=Xf=\mathcal L_Xf. \] For \(\mathrm{d}f\), both sides equal \(\mathrm{d}(Xf)\), using \(\mathrm{d}^2=0\) and the fact that Lie derivative commutes with \(\mathrm{d}\).
Commutator identities
Proposition. On forms, \[ [\mathcal L_X,\mathrm{d}]=0,\qquad [\mathcal L_X,\iota_Y]=\iota_{[X,Y]},\qquad [\mathcal L_X,\mathcal L_Y]=\mathcal L_{[X,Y]}. \]
These relations form the Cartan calculus. They encode how flow, differentiation, and contraction interact and are indispensable in symplectic geometry and characteristic-class theory.
Symmetries and conserved quantities
Example. A vector field \(X\) is Killing for a Riemannian metric \(g\) if \(\mathcal L_Xg=0\); its flow consists of local isometries. If \(\omega\) is a closed symplectic form and \(\mathcal L_X\omega=0\), Cartan's formula gives \[ \mathrm{d}(\iota_X\omega)=0. \] Locally \(\iota_X\omega=-\mathrm{d}H\), so the symmetry field is Hamiltonian and \(H\) is its local conserved quantity.