31. The connection map and covariant derivatives
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.
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