Leibniz rule for a connection

Let $\mathbb K=\mathbb R$ or $\mathbb C$ , let $E\to M$ be a smooth $\mathbb K$ -vector bundle, and consider an operation $(X,s)\mapsto\nabla_Xs$ on smooth vector fields and smooth sections, with values in smooth sections. Assume it is additive in each variable, $\mathbb K$ -linear in the section variable, and $C^\infty(M;\mathbb R)$ -linear in the real vector field $X$ . For $f\in C^\infty(M;\mathbb K)$ , the identity below is the additional axiom that defines a connection.

Definition. The Leibniz rule (product rule) for $\nabla$ is the identity

\nabla_X (f s) \;=\; X(f)\,s \;+\; f\,\nabla_X s.

Interpretation

This rule encodes that $\nabla$ differentiates sections “like a derivation” in the section slot, while remaining $C^\infty(M)$ -linear in the vector field slot.

Equivalent characterizations

Equivalently, using the 1-form $df$ defined by the exterior derivative of $f$ , one can write

\nabla(fs)=df\otimes s + f\,\nabla s

as an identity in $\Gamma(T^*M\otimes E)$ .

Examples

Trivial connection. On $E=M\times\mathbb R^r$ with $\nabla_X s=X(s)$ , the formula reduces to the usual product rule for differentiating a product of a scalar function and a vector-valued function.
Constant scalars. If $f$ is constant, then $X(f)=0$ and the rule becomes $\nabla_X(fs)=f\,\nabla_X s$ , expressing homogeneity over $\mathbb K$ -constants in the section argument. Additivity is part of the separately assumed bilinearity.
Local frame computation. In a local frame, writing $s=\sum_i s^i e_i$ and using the connection matrix $A$ , the rule is reflected in the identity $\nabla(s^i e_i)=ds^i\otimes e_i + s^i \nabla e_i$ .