Connections and Curvature Notes

where Dvai is the directional derivative of the function ai in the direction v. Consequently we know when a vector field does not change along a curve γ: Dγ̇X = 0. Covariant derivatives generalize the directional derivatives allowing us to differentiate vector fields on arbitrary manifolds and, more generally, sections of arbitrary vector bundles. Definition 1.1 (Covariant derivative of sections of a vector bundle). Let π : E →M be a vector bundle. A covariant derivative (also knows as a connection) is an R-bilinear map ∇ : Γ(TM)× Γ(E)→ Γ(E), (X, s) 7→ ∇Xs such that (1) ∇fXs = f∇Xs (2) ∇X(fs) = X(f) · s+ f∇Xs. for all f ∈ C∞(M), all X ∈ Γ(TM), and all s ∈ Γ(E). Example 1.2. Let U ⊂ R be an open set and E = TU → U the tangent bundle. Define a connection D on TU → U by DX( ∑ ai ∂ ∂xi ) = ∑ X(ai) ∂ ∂xi .