Covariant derivatives on vector bundles

A smooth function with values in ℝk on a manifold M can be viewed as a section of the trivial vector bundle M × ℝk. The theory of connections is an attempt to generalize the notion of directional derivative of (real or vector-valued) functions to sections in vector bundles.

Let π : E → M be a vector bundle. We are interested in operators which assign to each smooth vector field X on M and smooth section σ of E another smooth section of E called the covariant derivative of σ with respect to X. Of course, we would like these operators to be ℝ-linear, tensorial in the first variable and to satisfy the Leibniz rule. Summarizing, we have:

Definition 5.1. A covariant derivative on E is an ℝ-linear operator ∇ : C∞(M) × Γ(E) → Γ(E) denoted by (X, σ) ↦ ∇Xσ such that for all f ∈ C∞(M), X ∈ χ(M), σ ∈ Γ(E) the following conditions are satisfied:

1. (i) (Tensoriality) ∇f Xσ = f∇Xσ.

2. (ii) (Leibniz rule) ∇X(fσ) = f∇Xσ + (∂Xf)σ.

The first condition simply says that given a section σ, the value of ∇Xσ at some p ∈ M depends on only the value of X at p (Proposition 2.3).