In this lecture we introduce the connection map $\kappa \colon TE \to E$ of a connection on a vector bundle $E$ and prove that $\kappa$ is a vector bundle morphism along $T M \to M$.

We then introduce covariant derivative operators via the formula

$$\nabla_X(s) := \kappa (Ds [X]).$$

We will prove next lecture that a covariant derivative operator actually determines a connection uniquely, thus completely our “trifecta” of connection-related objects.