Lie derivative

Let $𝑣 𝑎 \in 𝔛 (𝑀)$ be a vector field generating the flow $𝜑$ . The Lie derivative

L 𝑣 : T 𝑝 𝑞 (𝑀) \to T 𝑝 𝑞 (𝑀)

is an $ℝ$ -linear map defined by¹ diff

L 𝑣 𝑇 𝑎 1 \dots 𝑎 𝑝 𝑏 1 \dots 𝑏 𝑞 = l i m 𝑡 \to 0 ((𝜑 𝑡) * 𝑇 𝑎 1 \dots 𝑎 𝑝 𝑏 1 \dots 𝑏 𝑞 - 𝑇 𝑎 1 \dots 𝑎 𝑝 𝑏 1 \dots 𝑏 𝑞 𝑡) = d d 𝑡 (𝜑 𝑡) * 𝑇 𝑎 1 \dots 𝑎 𝑝 𝑏 1 \dots 𝑏 𝑞 ∣ 𝑡 = 0 .

It is also possible to define the Lie derivative along a vector $𝑣 𝑎 \in 𝑇 𝑝 𝑀$ at a single point.

Algebraic properties

Agreement with other derivatives: $L 𝑣 𝑓 = 𝑣 (𝑓) = d 𝑓 𝑎 𝑣 𝑎 = \nabla 𝑣 𝑓$ .
Leibniz rule: $L 𝑣 (𝑆 \otimes 𝑇) = (L 𝑣 𝑆) \otimes 𝑇 + 𝑆 \otimes (L 𝑣 𝑇)$ .
Commutes with contraction: $L 𝑣 (𝑇 𝑎 1 \dots 𝑐 \dots 𝑎 𝑝 𝑏 1 \dots 𝑐 \dots 𝑏 𝑞) = L 𝑣 (𝑇 𝑎 1 \dots 𝑐 \dots 𝑎 𝑝 𝑏 1 \dots 𝑐 \dots 𝑏 𝑞)$ .

For any affine connexion $\nabla$ , we have

L 𝑣 𝑇 𝑎 1 \dots 𝑎 𝑝 𝑏 1 \dots 𝑏 𝑞 = 𝑣 𝑐 \nabla 𝑐 𝑇 𝑎 1 \dots 𝑎 𝑝 𝑏 1 \dots 𝑏 𝑞 - 𝑝 \sum 𝑖 = 1 𝑇 𝑎 1 \dots 𝑐 𝑖 \dots 𝑎 𝑝 𝑏 1 \dots 𝑏 𝑞 \nabla 𝑐 𝑖 𝑣 𝑎 𝑖 + 𝑞 \sum 𝑖 = 1 𝑇 𝑎 1 \dots 𝑎 𝑝 𝑏 1 \dots 𝑐 𝑖 \dots 𝑏 𝑞 \nabla 𝑏 𝑖 𝑣 𝑐 𝑖

Proof

proof