Definition. Covariant Derivative (Linear Connection) [math-HCIJ]

2026-02-04 · Lîm Tsú-thuàn

Let E→ME \to M be a vector bundle. A covariant derivative on EE is a K\mathbb{K}-linear map

∇:C∞(E)→C∞(T∗M⊗E)\nabla : C^\infty(E) \to C^\infty(T^*M \otimes E)

such that, for all f∈C∞(M)f \in C^\infty(M) and all u∈C∞(E)u \in C^\infty(E), we have

∇(fu)=df⊗u+f∇u\nabla(fu) = df \otimes u + f \nabla u

where C∞(E)C^\infty(E) denotes the space of smooth sections of EE over MM.

Remind that

Hom(T∗M,E)≃C∞(T∗M⊗E)\text{Hom}(T^*M, E) \simeq C^\infty(T^*M \otimes E)

Therefore, ∇u\nabla u has a more traditional view

∇u:Vect(M)→C∞(E)X↦∇Xu\begin{aligned} &\nabla u : \text{Vect}(M) \to C^\infty(E) \\ &X \mapsto \nabla_X u \end{aligned}