I am studying differential geometry for the n-th time re-reading the concepts and making sense on formulas and concepts. A fundamental operator is the connection, especially the Levi-Civita connection, whose properties are below.
The author Oxford Professor Jason Lotay utilizes the domain-image notation $\nabla : \Gamma(TM) \times \Gamma(TM) \to \Gamma(TM)$. Due my ignorance, I understand sections are a somewhat family-like notation for maps applied on vector fields. However, until this point of my academic life, all operations were done locally, not respective to different points $p$ and $q$ in space, except for distances in metric spaces. Specially for covariant derivative $\nabla$, further read talks about parallel transport along given curve $\alpha(t)$, which we can choose as the geodesic curve $\gamma(t)$
My quesiton is: is it or is it not a local operator? If not, how can I make sense of it on a manifold which is not an immersion or a submersion, and notion of outer-manifold is not applicable?