Local chart connection

Suppose $(U, \varphi)$ is a local chart of $M$. Given a vector field $\xi$ we can express it in the local chart as $\xi=\sum_i f_i \partial{u_i}$. Given other vector, say $\eta$, we can define in $U$ the expression

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 $(U,\varphi)$.

Moreover, this operator is unique: it is the only that satisfies

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 $C^a_{nm}$ like in the entry contorsion. The Christoffel symbols $\Gamma_{ij}^k$, 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 $C_{mn}^a$ expressed in the basis $\{(dx_i)_m (dx_j)_n (\partial x_k)^a\}_{ijk}$.

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

• For vectors

• For 1-forms

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

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Author of the notes: Antonio J. Pan-Collantes

antonio.pan@uca.es