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Covariant derivative of a section

The derivative of a vector bundle section along a vector field as defined by a connection.

Let E→ME\to M be a vector bundle with a ∇\nabla. For a XX on MM and a section s∈Γ(E)s\in\Gamma(E), the connection produces another section ∇Xs\nabla_X s.

Definition. The covariant derivative of ss along XX is the section

∇Xs∈Γ(E)\nabla_X s \in \Gamma(E)

defined by the connection’s bilinear map ∇:X(M)×Γ(E)→Γ(E)\nabla:\mathfrak X(M)\times\Gamma(E)\to\Gamma(E). It is characterized by:

  • C∞(M)C^\infty(M)-linearity in XX: ∇fXs=f ∇Xs\nabla_{fX}s=f\,\nabla_X s,
  • R\mathbb R-linearity in ss, and
  • the Leibniz rule in the section slot (see ).
Equivalent characterizations

Equivalently, viewing ∇s\nabla s as a T∗M⊗ET^*M\otimes E-valued object, one has (∇Xs)(p)=(∇s)(p)(Xp)(\nabla_X s)(p)=(\nabla s)(p)(X_p) by contraction.

Examples
  1. Trivial bundle: ordinary directional derivative. For E=M×RrE=M\times\mathbb R^r with the trivial connection, ∇Xs\nabla_X s is just the usual derivative of the vector-valued function ss in the direction XX.
  2. Tangent bundle: covariant derivative of vector fields. For a connection on TMTM, ∇XY\nabla_X Y is the covariant derivative of the vector field YY along XX, recovering the classical Christoffel-symbol formula in coordinates.
  3. Line bundle with connection 1-form. In a local trivialization of a complex line bundle with connection form AA, if s=f es=f\,e for a local frame ee, then ∇Xs=(Xf+A(X) f) e\nabla_X s = (Xf + A(X)\,f)\,e.