Local chart connection

Suppose $(U, φ)$ is a local chart of $M$ . Given a vector field $ξ$ we can express it in the local chart as $ξ = \sum_{i} f_{i} \partial u_{i}$ . Given other vector, say $η$ , we can define in $U$ the expression

\nabla_{η} ξ = \sum_{i} η (f_{i}) \partial u_{i}

Implicitly, what we are doing is assume a geometry in which the $\partial u_{i}$ are constant. It could be shown that $\nabla$ define a covariant derivative operator restricted to $U$ , that applies to any tensor field, not only vectors. It is called the local chart connection or coordinate connection for $(U, φ)$ .
Moreover, this operator is unique: it is the only that satisfies

\nabla (\partial u_{i}) = 0

When there is no place to confusion, we will denote it by $\partial$ .

How is this related to Christoffel symbols? Think of any connection $\nabla$ . In a local chart, we have the trivial connection $\partial$ , and the expression $\nabla - \partial$ defines a set of tensor fields $C_{n m}^{a}$ like in the entry contorsion. The Christoffel symbols $Γ_{i j}^{k}$ , defined traditionally as the components of $\nabla_{\partial_{x_{i}}} \partial x_{j}$ respect to the basis element $\partial x_{k}$ , multiplied by -1 coincide with the components of $C_{m n}^{a}$ expressed in the basis ${(d x_{i})_{m} (d x_{j})_{n} (\partial x_{k})^{a}}_{i j k}$ .

So, in local charts, a connection can be expressed:

For vectors

\nabla_{l} V^{m} = \frac{\partial V^{m}}{\partial x^{l}} + Γ_{k l}^{m} V^{k}

For 1-forms

\nabla_{l} ω_{m} = \frac{\partial ω_{m}}{\partial x^{l}} - Γ_{m l}^{k} ω_{k}

For general tensors, the expression could be deduced from contorsion.