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A connection (or covariant derivative) ∇ provides a way to differentiate vector fields along curves on a manifold — something that requires additional structure beyond the smooth structure because there is no canonical way to compare tangent vectors at different points. The covariant derivative ∇_X Y measures how Y changes as you move along X, and it is determined in coordinates by the Christoffel symbols Γᵏᵢⱼ. Unlike the Lie bracket, the covariant derivative is tensorial in its first argument, making it the right tool for defining parallel transport and curvature.
On ℝⁿ with standard coordinates, differentiating a vector field Y = Yⁱ eᵢ in the direction X is straightforward: you differentiate the components X(Yⁱ). This works because the standard basis vectors eᵢ are constant — they do not change from point to point. On a curved manifold, the coordinate basis vectors ∂/∂xⁱ vary from chart to chart, and there is no canonical notion of "constant vector field." A connection provides the missing ingredient: it specifies how to transport vectors infinitesimally from one tangent space to a nearby one.
A covariant derivative (affine connection) is an operation ∇ : 𝔛(M) × 𝔛(M) → 𝔛(M) satisfying: (1) ∇_{fX+gY} Z = f∇_X Z + g∇_Y Z (C∞(M)-linear in the first argument), (2) ∇_X(Y+Z) = ∇_X Y + ∇_X Z (additive in the second argument), and (3) ∇_X(fY) = X(f)Y + f∇_X Y (Leibniz rule in the second argument). In local coordinates, the connection is specified by its Christoffel symbols Γᵏᵢⱼ, defined by ∇_{∂/∂xⁱ}(∂/∂xʲ) = Γᵏᵢⱼ ∂/∂xᵏ. The covariant derivative of Y along X is then (∇_X Y)ᵏ = X(Yᵏ) + Γᵏᵢⱼ Xⁱ Yʲ — the first term differentiates components, the second corrects for the changing basis.
The C∞(M)-linearity in the first argument is the crucial property. It means (∇_X Y)_p depends only on the vector X_p ∈ TpM, not on how X extends away from p. This is what makes ∇ tensorial in its first argument — you can evaluate ∇_v Y for a single tangent vector v, which the Lie bracket cannot do. The Leibniz rule in the second argument means ∇_X Y does depend on the behavior of Y along the direction X (it sees the first derivative of Y), making it genuinely differential.
The torsion of a connection is T(X,Y) = ∇_X Y - ∇_Y X - [X,Y], measuring the antisymmetric part of ∇ beyond what the Lie bracket accounts for. A torsion-free connection satisfies ∇_X Y - ∇_Y X = [X,Y]. On a Riemannian manifold, the Levi-Civita connection is the unique torsion-free, metric-compatible connection. But connections exist independently of any metric — they are a more primitive notion than Riemannian geometry, and they are the natural structure on general vector bundles, principal bundles, and gauge theories in physics.